New results in the theory of multivalued mappings2

N E W R E S U L T S IN T H E T H E O R Y O F M U L T I V A L U E D M A P P I N G S . II. A N A L Y S I S AND A P P L I C A T I O N S

B. D. Gel'man and V. V. Obukhovskii

UDC 517.988+515.126.83+ 517.911.5+517.977

This is the second part of a survey by Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkins, and V. V. Obukhovskii on new results in the field of analysis of multivalued mappings and their applications during the 1980 's.

INTRODUCTION This paper is the second part of a survey by Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkins, and V. V. Obukhovskii ("New results in the theory of multivalued mappings. I. Topological characteristics and solvability of operator relations," Itogi Nauki i Tekhn., Ser. Mat. Anal., Vol. 25, 123-197 (1987)). It is devoted to fundamental results obtained in the field of analysis of multivalued mappings and their applications over the period 1981-1986. An appendix by V. V. Obukhovskii presents a brief survey of research in recent years (including research that touches on the topological characteristics and fixed points of multivalued mappings). The survey is limited to the most important results or results close to the interests of the authors, since the volume of publications on this subject in recent years is overwhelming. The references in the text refer to the bibliography in the first part of the survey; an additional bibliography is provided here.

CHAPTER 1 ANALYSIS OF MULTIVALUED MAPPINGS

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Spaces of Closed Subsets

Since any multivalued mapping must be treated as a mapping into a family of subsets of some set that is equipped with some structure or other, the study of such structures is of great interest. The greatest interest has been focused on the ordinary set of closed subsets of a topological space. Before proceeding, we will recall some notation. Let X be a topological space; P(X) denotes the set of all nonempty subsets of X. Then C(X) = {A E P(X): A is closed}; K(X) = {A E P(X): A is compact}. If X is a subset of a topological vector space, then Pv(X) is the set of all of its nonempty convex subsets; Cv(X) = C(X) f~ Pv(X); Kv(X) = K(X) f~ Pv(X). The set C(X) can be converted into a topological space in various ways. The following topologies are the most commonly used: the exponential topology, the upper semifinite topology, and the lower semifinite topology (see [45]). We denote the set C(X), equipped with the exponential topology, by ExPc(X). On the sets K(X), Cv(X), and Kv(X) the exponential topology induces the topologies Expi<(X), Expcv(X ) and ExpK~(X). We denote the upper and lower semifinlte topologies on C(X) by xc(X) and Xc(X), respectively. On K(X), Cv(X), and Kv(X), they induce XK(X), Xcv(X ), XKv(X) )~K(X), and Xcv(X), XKv(X), respectively. The homotopic properties of spaces of subsets were studied in [25-27]. In particular, the following propositions were proved there:

Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 29, pp. 107-159, 1991.

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0090-4104D3/6402-0854~$12.50 9

Plenum Publishing Corporation

1.1.1. T H E O R E M . Let X be a subset of a Banach space. Then the spaces ExPKv(X) and X are of the same homotopic type. 1.1.2. T H E O R E M . Let X be a subset of a Banach space; then, for any point xo E X, the two homotopy groups %(X, Xo) and %(kKv(X), x0), n > 0, are isomorphic, and the isomorphism is induced by an imbedding of (X, xo) in

(X~:v(X), x0). 1.1.3. T H E O R E M . Let X be an open linearly connected subset of a closed convex set in a Banach space. Then, for any n _> 0, the two homotopy groups %(X) and %(XKv(X)) are isomorphic, and the isomorphism is induced by an imbedcling of X in xKv(X). 1.1.4. T H E O R E M . Let X be a Hausdorff topological space9 If X is linearly connected and locally contractible, the space Exp,(X) is linearly connected and the homotopy groups %(ExpK(X)) are trivial for all n _> 0. 1.1.5. T H E O R E M . Let X be a topological space; then • and kK(X) have point homotopies. In [15] S. M. Aseev defined the notion of a quasilinear space, whose axioms characterize the properties of spaces of closed subsets of normed spaces9 Following this paper, we present the following fundamental definitions. 1.1.6. Definition. A set X is said to be a quasilinear space if a partial ordering _< is defined on it, and algebraic addition and multiplication for real numbers are defined so that for any x, y, z, u C X and c~,/3 E R we have: 1) x _< x; 2) x_< y a n d y < z i m p l y x _ < z ; 3 ) x _ < y a n d y _ < x i m p l y x = y ; 4 ) x + y = y + x ; 5 ) • + ( y + z) = ( x + y ) + z ; 6 ) 30 E X: x + 0 = x; 7) ot(/3x) = (oe/3)x; 8) c~(x + y) = eex + o~y; 9) l ' x = x; 10) 0"x = 0; 11) (c~ + /3)x _< cex + /3x; 12) x _< y a n d z -< u i m p l y t h a t x + z _.< y + u; 13) i f x _< y, thenc~x _< o~y. 1.1.7. De• Let X be a quasilinear space. A real function II IIx: x --, R is called a norm in X if the following conditions are satisfied: 14)Ilxllx > 0 i f x # 0; 15)Ilx + yllx -< Ilxllx + Ilyllx; 1 6 ) I I ~ x l l x -- I~1 II~llx; 1 7 ) i f x < y, then ilX]lx -< Ilyllx; 18)if, for any ~ > 0, there exists an element x e E X such that x < y + x e and IIx llx -< ~, then x _< y. A quasilinear space X with a norm defined in it is called a normed quasilinear space. If X is a normed quasilinear space, then we can define a function hx: X • X + R so that h v ix, !/) =inf{rj~0 : x~y+a~", g~x+a2 ~, II~,qlx~
A, BtC (E), A+B={a+b [acA, b~B}. We define a partial ordering in C(E) by set inclusion. Then C(E) is a quasilinear space. If we define a norm in C(E) with the formula ][ A IIc { e ) = sup IIa tl,

a~A

then C(E) is a normed quasilinear space. We should note that K(E), Cv(E), and Kv(E) with similarly defined operations are quasilinear spaces. Note that it is possible to give examples of quasilinear spaces that cannot be realized as subspaces of a quasilinear space of subsets of some normed space. The conditions for such a realization are given in a proposition proved by B. D. Gel'man and E. V. Chernodubov. 1.1.9. T H E O R E M . Let X be a normed quasilinear space, and let E be the set of minimal elements in it. Assume that the following conditions are satisfied in X: a) for any elements x, y, and z such that x + y ~ z there exist sequences {Xn}n=l ~, {Yn}n=l ~, and {Zn}n=l '= such that for any n: x n _< x, Yn -< Y, zn -< z, and x n + Yn -< zn, where Znh -+ z; b) if 9 x x g y, then there exists an xo E E such that x o _< x, Xo _< y. Then the set E is a normed hnear space and there exists a mapping 7r: X ---, C(E) such that: 1) 7r is injective; 2) 7r(x + y) D 7r(x) = 7r(y), Vx, y E X; ; 3) 7r(kx) = Mr(x), V~, E R, xEX. Aseev [15] also develops analysis for quasilinear spaces and proves many facts similar to those of classical analysis (the H a h n - B a n a c h theorem, etc.). Spaces of subsets are also considered in [23, 69, 409, 607, 628,633, 967, 1107, 1192]. The survey [4] is especially noteworthy; it contains a systematic exposition of the different properties of topologies in C(X), where X is a metric space, and provides a comparison of convergence in different topologies.

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w Continuity of Multivalued Mappings. Operations On Multivalued Mappings We begin by recalling that a multivalued mapping (m-mapping) of a set X into a set Y is a correspondence F: X --, P(Y). If X and Y are topological spaces, we say that an m-mapping F: X ~ P(Y) is: a) upper semicontinuous if F + - I ( v ) = {x 6 X : F(x) C V} is open in X for any open V C Y; b) lower semicontinuous if F _ - I(V) = {x E X: F(x) N V ~ ~} is open in X for any open V C Y; c) continuous if F is both upper and lower semicontinuous; d) closed if its graph PF = {(x, y) 6 X X Y : y E F(x)} is closed in X • Y. These types of continuity of m-mappings and their properties were studied in [45, 47]. A new characterization of upper and lower semicontinuity for m-mappings was given in [993]. Various conditions for semicontinuity and continuity of m-mappings in a metric space are given in [45, 47, 248]. Research on the topological structures of upper semicontinuous mappings is the subject of [737, 738]. In [1160] there is a unified classification scheme for various types of semicontinuity of m-mappings that is based on using the notions of semiclosed operators, generalized topologies, etc. The notion of generalized upper semicontinuity is introduced, and an analytic criterion for this property is given. Anisiu [358] describes the relationship between upper semicontinuity, lower semicontinuity, and continuity in the Hausdorff metric for m-mappings with closed bounded images in a metric space. Certain properties of lower semicontinuous m-mappings were studied in [685]. In particular, this paper presents conditions under which m-mappings that are defined in a space with two topologies and are upper semicontinuous in one of them have a residual set of points that is lower semicontinuous with respect to the second topology. It also presents a theorem on single valuedness of monotonic lower semicontinuous m-mappings. Theorems on the existence of "sufficiently large" subspaces in domains on which upper or lower semicontinuous mmappings are continuous are considered in [514, 761,886, 1124]. A number of papers have considered new types of continuity for m-mappings. Aubin and Cellina [370] present a detailed description of the class of upper hemicontinuous m-mappings, which has important applications. 1.2.1. Definition. Let X and Y be locally convex spaces; an m-mapping F: X --> P(Y) is said to be upper hemicontinuous if, for any p E Y* the function ~: X --, ( - a o , + ~ ) , q5 (x) = c (F (x) ; p) = sup:j~>/

is upper semicontinuous (in the single-valued sense). The properties of m-mappings that are upper semicontinuous in the sense of Ceasan are described by Glodde and Niepage [628]. The authors of [391, 414, 446, 447, 448, 575, 576, 715,923-928, 960, 961, 1011, 1013-1016, 1018, 1020-1022] introduce a whole series of different types of continuity for m-mappings and investigate the relationships between them. Various aspects of the notion of continuity for m-mappings are considered in [597, 628, 1216, 1231, 357]. The property of uniform lower semicontinuity for a family of m-mappings was studied in [934]. Continuity of linear m-mappings is the subject of [389, 390]: Let X and Y be normed spaces. An m-mapping F: X ---, P(Y) is said to be linear if: a) F(x + x') -- F(x) + F(x'), Yx, x' 6 X; b) F(ccx) = ~F(x), Yx 6 X, ot ~ 0; c) 0y E F(0x). If F is a linear m-mapping, then: 1) F is upper (lower) semicontinuous on X if and only if it is upper (lower) semicontinuous at 0x; 2) if F is upper semicontinuous at 0x, then F is lower semicontinuous at 0x; 3) if F is upper semicontinuous at 0x, then F(0X) = {0y} or F~XX) = Y; 4) F is lower semicontinuous if and only if the induced linear mapping f: X --, Y/F(0x) is continuous. The authors of [1050, 1051, 1052] introduce the notions of linear m-mappings of vector spaces, multimatrices, and the inverse matrices of such mappings; they investigate conditions for continuity of linear m-mappings. Properties of mmappings inverse to a given m-mapping are studied in [1053]. Nicodem [944] studied conditions under which additive m-mappings defined on a set of nonempty convex subsets of metric linear spaces are continuous in the Hausdorff metric. An m-mapping F of topological vector spaces X and Y is said to be convex with respect to midpoints if

F(Ix

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- ,~ -1~ x ,\} ~ - F1 ( x ) ~ - T F~ ( tx '

),

Vx, x ' E X .

It was shown in [1176] that if an m-mapping is convex with respect to midpoints and lower semicontinuous at some point, it is lower serniconfinuous everywhere in int(dom F), where dora F = {x E X: F(x) ~ q~}. The structure of closed m-mappings of Banach spaces with convex graphs was described by Le Van Hot [819]. In particular, it was shown that if at least one image F(xo) of such a mapping F is bounded, then F is of the tbrm F(x) = F(0) + t(x), where t is a linear one-to-one mapping. Closed m-mappings with convex graphs were also considered in [370, 1057]. In particular, it was shown that such mmappings are lower semicontinuous inside their domains. The property of continuity for m-mappings whose graphs are the union of a finite number of polygonal convex sets were studied by Robinson in [1062J. This paper also considers applications to mathematical programming. Papers by Ledyaev [151] and Rockafellar [1065] consider m-mappings whose images are sets given by a system of constraints containing equations and inequalities. They also derive conditions under which the indicated mappings satisfy the Lipshitz condition in the Hausdorff metric. Applications to problems of parametric nonlinear programming are given. Sufficient conditions for lower semiconfinuity of the set of solutions to a system of convex inequalities are given by Klatte in

[782]. Let X and Y be topological spaces; for a function f: X x Y --, R we define m-mappings V, E, and Q from X into Y ' V ( x ) = {y E Y : f ( x , y) = 0}; E(x) = {y E Y - y is an extremum of the function f(x, ")}; Q (x) = V(x)\E(x). Conditions for nonemptiness and closure of the images are given by Ricceri in [1055], along with conditions for lower semicontinuity of the m-mapping Q. It is well known that upper semicontinuous and closed m-mappings are most frequently encountered in applications in game theory, economics, and optimal control theory. However, Dolecki [551] considers the role of lower semicontinuity in the theory of optimality, and describes its connection with the notions of minima and extrema. A number of papers consider various operations on multivalued mappings and their properties. Various continuity properties for such operations were described in [45, 47, 235]. Aliev [3] studied closure for certain m-mappings that appear in differential and integral inclusions. The following basic results were obtained there: 1.2.2. T H E O R E M . Let X and Y be T 1-spaces, and F: X --, P(Y) be an arbitrary m-mapping. If F: X --, C(Y), r x = Y--~U--), where .;r (v) is a basis of neighborhoods of x, then r is closed and F is closed if and only ff F = F. L'(-; ~ , x j

1.2.3. T H E O R E M . If F: X --, C(Y) is closed, then {x E X: B c Fx} is closed for any B C Y, which is equivalent toPx = " i"(U) for a n y x E X. 1.2.4. T H E O R E M . Let Y be a topological vector space, let F: X --) P(Y) be an arbitrary m-mapping, and P : X -) CV(Y),Px = c o F ( U ) . Then F is closed and Px = coF(U) for a n y x E X. The closure of the composite of an m-mapping and a linear operator is investigated, along with Volterra type moperators. For an m-mapping F: R x R n --, Kv(Rn), let

KF(~, x-)~ ~'~ F! ~>0

~(,v) =0

coF(t, B(x, 6 ) \ N ) ,

where # is the Lebesgue measure on R. Operations of this sort were first considered by Filippov (see, for example, [2991) for reduction of differential equations with continuous right sides to differential inclusions. Some properties of the m-mapping KF were described in [733, 734]. 1.2.5, Definition. Let X and Y be finite-dimensional spaces, Z = X x Y; for an m-mapping F: X --, P(Y), the mmapping F*: Y* x I"F --, P(X*),

F* (y*, z) ={x*~X* : ( --x*, V*)GK~* (z)}, where PF is the graph of F and KF*(Z) is the cone conjugate to the cone of tangents at z, is said to be locally conjugate to F. Makhmudov [164] computed locally conjugate mappings for cross-sections, sums, direct products of m-mappings, scalar products of m-mappings, and m-mappings inverse to given m-mappings.

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Pointwise upper limits for generalized sequences of upper semicontinuous m-mappings were considered by B~nzaru [384], and conditions for upper semicontinuity of such limits are given. Theorems on preservation of continuity upon passage to the limit in sequences of m-mappings are given in [500, 522], along with theorems on quasicontinuity and upper and lower semicontinuity. Different definitions for sequences of m-mappings are also considered in [597, 788, 789, 1197]. The topological properties of "limit" m-mappings are studied in [788, 789], along with applications to optimization problems associated with certain problems in game theory and mathematical economics. An analog of the classical theorem on the existence of pointwise convergent subsequences in sequences of m-mappings is given in [523]. The notion of the analytic limit of m-functions at a point is considered in [1086, 1087]. The property of analytic convergence of sequences of m-mappings is described in [1088, 1089]. Let F be an m-mapping of topological spaces X and Y, and let f: X • Y --, R be some function. The function ,p:

X-,R, (p(x)= max f (x, V)

is called the maximum function (see [45, 47]). The problem of minimizing the maximum function was studied in [41], where necessary conditions for the minimax were obtained. This is also the subject of [38, 167, 246], which also investigated differentiability of maximum functions with respect to directions, and derived formulas for computing derivatives. Let X and T be Hausdorff topological spaces, and let Y be a Hausdorff topological vector space ordered by a convex closed cone that contains no lines. Let r be an m-mapping from T into X, f: X -- Y, and let Mr(t ) be the set of minimal elements of the set f(I'(t)). Conditions under which the t--- Mf(t) is lower semicontinuous were derived in [1159]. Lyapin [158] presented conditions on boundedness, semicontinuity, and continuity of m-superposition-operators (see [45, 47]) without assumptions on the convexity of the functions they generate. Conditions for complete continuity of the product of a linear completely continuous operator and an m-superposition operator are given.

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Properties of Multivalued Mappings

The topology on the space of m-mappings of a topological space that extends the topology of uniform convergence to compact spaces of single-valued mappings was introduced by Gupta in [629]. A number of Ascoli-type theorems on compactness of subsets of continuous m-mappings were generalized by Morales in [896]. New criteria for weak compactness in the space of bounded m-mappings with convex compact images in a reflexive Banach space were obtained in [461]. Applications of the results obtained were given for optimization problems and for theorems on existence of best approximations in Banach spaces. The following analogy of the classical principle of uniform boundedness was proved for m-mappings in [936]: Let X and Y be real normed spaces; an m-mapping F from X into Y is said to be convex if ~zF(x) + (1--,co) F (y) c F (~x+ (1--c~) V) for any x, y E X and 0 < a < 1. The norm 3,(F) is defined as (F)~

sup

inf ]iY[I.

Ilxll~
Let {Fj}jE J be the family of convex upper semicontinuous m-mappings from X into Y such that U Fs (x) is bounded in Y

jilt

for all x E X and there exist a, b >_ 0 such that Fj(ax) C ac~Fj(x) + By(0, b) is satisfied for all x E Bx(0 , 1) and j E J, 0_< a _< 1. Then sup '/(F j) < oo . Variants of the generalized principle of uniform boundedness to the case of m-mappings were also considered in [216, 218, 933]. An analog of the Hahn-Banach theorem for m-mappings was proved in [475]. Theorems generalizing the theorem on closure of the graph of open mappings were proved for m-mappings of quasimetric spaces in [765]. 858

A form of the theorem on open mappings for m-mappings of Banach spaces was considered in [602], which also presents corollaries concerning the problem of local controllability of systems described by differential inclusions. The following theorem on representation of m-mappings was proved in [1232]. Let X be a complete metric space; an m-mapping F: [a, b] --, K(X) is absolutely continuous in the Hausdorff metric if and only if there exists a compact metric space U and a continuous mapping f: [a, b] • U --, X that is absolutely continuous in the first argument and such that

F(t) = f ( t , U) for all t E [a, b]. Integral representations of m-mappings were studied by Papageorgiou in [976]. Let X be a normed space, and let K C X be some convex closed cone. An m-mapping F: K -+ P(K) is said to be superlinear if: 1) F(0) = {0}; 2) F(ax) = o~F(x), Vx E K, a > 0; 3) F(x~ + x2) D F(xl) + F(x2) , vxlx 2 E K. Makazhanova [160] studied conditions affecting the existence of characteristic compacta for superlinear m-mappings F, i.e., nonzero convex compact sets N C K such that F(I~) = XI] for some X > 0. The papers [561, 971] introduce fuzzy m-mappings, i.e., m-mappings in fuzzy topological spaces. They define the notions of upper and lower semicontinuity, measurable fuzzy m-mappings, the integral of fuzzy m-mappings, and they establish a number of properties and indicate applications. Let X be a compactum, and let F: X --, C(X) be some m-mapping. Its trajectory is the sequence {Xn}n=o= C X such that xn+ 1 E F(xn), n = 0, 1. . . . . Investigation of the topological properties of trajectories and magistracies, i.e., the smallest closed sets to which any trajectory of the m-mapping converge, is the subject of [95, 96, 97]. Semigroups of m-mappings are studied in [1140, 1t41]. The notion of almost periodic m-mappings is introduced and investigated in [385-388]. Arestov [12] considered the problem of the best restoration of the values of a multivalued operator on a class of elements by means of a given set of single-valued operators. Let S be a set, let [S]2 be the set of all of its two-element subsets, and let the m-mapping F: IS] 2 --, P(S) be such that a n F(a) = 2~ for all a E [S] 2. A subset H C S is said to be free for F if, for any x, y, z E H, we have x ff F({y, z}). Williams obtained conditions under which F has relatively large free sets in [1213]. A multivalued function F: C ~ K(C) is said to be analytic in G if the set {(z, X): X C Gz ff F(X)} is pseudoconvex in C;. Analytic m-functions were introduced in 1934. Interest in them has increased in recent years in connection with applications to problems in functional analysis. The properties of analytic m-functions are the subject of [375-377, 584, 10401042, 1138, 1139]. In [499] there is an intuitive method for describing m-mappings with decision diagrams that consist of trees in which arguments are assigned to the vertices and their values are assigned to the branches. Various properties of m-mappings were also investigated in [71, 75, 82, 153, 154, 161, 170, 199, 200, 216, 315, 349, 378, 436, 478, 552, 553,560, 597, 619, 620, 624, 709, 743,760, 770, 792, 820, 834, 836, 838, 867, 906, 907, 945, 946, 1029, 1068, 1084, 1089, 1129].

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Continuous Sections of Multivalued Mappings Let X and Y be sets, and let F: X -> P(Y) be an m-mapping. 1.4,1. Definition. We call a single-valued mapping f: X ~ Y such that

f(x) ~F(x) for all x E X a section of the m-mapping F. If X and Y are topological spaces and the mapping f is continuous, we say that f is a continuous section of F. Smooth, Lipshitz, and other classes of sections are naturally defined. In Sec. 1.7 we will consider measurable sections of mmappings. The existence problem for continuous sections is very interesting and has many applications in general topology and other areas of modern mathematics (see the detailed bibliography in [45]). One of the fundamental existence theorems for continuous sections is Michael's classical theorem (see [45]). Michael's recent papers (in conjunction with other mathematicians) [878-881] have been concerned with the existence problem for continuous sections of lower semicontinuous m-mappings. We present the following proposition as an example.

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1.4.2. THEOREM. Let X be a paracompact topological space, Z C X, dimxZ _< 0, let W C X be a countable subset, and let Y be a Banach space. Let F: X ~ P(Y) be a lower semicontinuous m-mapping such that F(x) E C(Y) for x W and F(x---) 6 Cv(Y) for x ~ Z. Then F has a continuous section. These papers are closely related to [511, 513, 680, 823, 853, 916]. In [502], Curtis generalizes Michael's classical theorem to metric spaces in which convex structures have been introduced. The relationship between the existence of continuous sections of m-mappings and the properties of topological spaces is studied in [392, 427, 856, 858, 905, 915,918,919]. For example, [858] establishes a connection (of the type described in Michael's theorem) between C-paracompactness and C-collective normality of spaces and the existence of sections, while [905] investigates the connection between the properties of z-strong normality and the existence of sections, etc. Much interest has recently been drawn to the study of m-mappings with values in the functional space of space of summable functions L 1. When these m-mappings are studied, the convexity condition is replaced by the condition of decomposability for the images, which is more natural because of the specific characteristics of the space L 1. 1.4.3. Definition. A set A C LI(I), where I is some interval, is said to be decomposable (or convex with respect to switching) if, for any functions fl, f2 E A and any measurable subsets 11, 12 C I, 11 U 12 = I, 11 ('1 12 = ~ we have x l f l + xizf2 E A, where Xh and XI2 are the characteristic functions of 11 and 12, respectively. Bogatyriev [34] and Fryszkowski [610] have proved analogs of Michael's theorem on existence of continuous sections for lower semicontinuous m-mappings from compact X to the space L 1 with decomposable images. These results have been applied to theorems on fixed points for m-mappings with decomposable images. This approach is taken in [34] to solvability of the Cauchy problem for differential inclusions. Further results in this area are considered in the Appendix. Kolesnikov [115-120] investigated the existence of continuous sections for continuous and lower semicontinuous mmappings with G,-diagonals in ordered spaces, expanded spaces, etc. We present the following propositions as examples. 1.4.4. THEOREM. Let X be a zero-dimensional collectively normal space, and let Y be a metric space. Then every lower semicontinuous m-mapping F: X ---, C(Y) has a continuous section. 1.4.5. THEOREM. Let X be a Baire space, and let Y be a complete metric space. Then every lower semicontinuous m-mapping F: X --, C(Y) has a continuous section on a set of type G~ that is everywhere dense in X. The following proposition was proved by Saint Raymond [ 1092]. 1.4.6. THEOREM. Let X be a metrizable compact space of dimension no less than n, and let Y be a Banach space. Suppose that the m-mapping F: X --, Cv(Y) is lower semicontinuous, each of its images F(x) contains zero, and each image is of dimension no less than n. Then F has a continuous section that does not vanish. A theorem on the existence of a continuous section that generalizes Brouder's theorem on sections of m-mappings with convex images and open preimages is proved by Pasicki in [986]. Gel'man [72] found necessary and sufficient conditions for existence of continuous sections of compact m-mappings with convex closed images (without assumptions concerning continuity) in terms of local structures. This paper, as well as McClendon's papers [869, 871], studies the existence problem for continuous sections of mmappings with nonconvex images (this situation is also considered in the Appendix). If F: X --, P(Y) is an m-mapping of a topological space X into a metric space (Y, p), then, for e > 0, a continuous mapping fe: X --, Y is said to be an e-section of F if

~(L(x),/~(x)) < ~ for every x 6 X. Similarly, if Y is a topological vector space and U C Y is a neighborhood of zero, then f: X --, Y is called a U-section of F if

f (x) ~F (x) + U for all x 6 X. The existence problem for this kind of approximate continuous section, which problem finds its origin in work of Michael, has recently been considered in the papers [46, 72, 527, 665-667, 674, 675, 680]. The relationship between local structures of m-mappings and the existence of continuous sections or e-sections was described in [46, 72]. Lipshitz sections of Lipshitz m-mappings were investigated in [314, 333, 554]. The existence problem for uniformly continuous sections of uniformly continuous m-mappings was studied in [267268].

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The interesting problem of the existence of smooth sections for m-mappings was considered in [14, 72]. Various other aspects of the existence problem for sections were considered in [94, 107, 138, 181,254, 255, 406, 408, 428,450, 515, 528, 582, 688, 721,853, 917, 1097, 1132, 1133, 1194, 1195, 1212].

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Continuous Approximations of Multivalued Mappings Let (X, Px) and (Y, py) be metric spaces. We define the metric p in the product space X • Y with the equation

9((x, y), (x', y'))=max{px(x, x'), py(y, y')}. 1.5.1. Definition. A continuous single-valued mapping fe: X --> Y, where e > 0; is said to be an e-approximation of an m-mapping F: X --> P(Y) if the graph Ffe of the mapping fe is located in an e-neighborhood of the graph F F of the mmapping F. Single-valued approximations are convenient and important instruments for investigating re:mappings, their topological characteristics, stationary points, etc. (See, for example, [44-47].) Among the theorems on existence of eapproximations we note the following result of Anichini, Conti, and Zecca [354]: 1.5.2. THEOREM. Let F be an upper semicontinuous m-mapping from [0, T] x R n into R n with nonempty compact contractible images. Then, on each compact topological polyhedron W C [0, T] x R n, the m-mapping F admits a unique t-approximation for any e > 0. This result was further developed in [355]. The existence of a single valued t-approximation for an upper semicontinuous m-mapping with decomposable images in L I was proved by Cellina, Colombo, and Fonda in [468]. The existence problem of single-valued approximations of m-mappings with nonconvex images was also considered in the survey [44]. For recent results concerning approximable m-mappings, see the Appendix. In addition to single-valued approximations, interest has been drawn to continuous multivalued approximations of mmappings. The following propositions were proved by Aseev in [13]: 1.5.3. THEOREM. Let X be a metric space; then an m-mapping F: X --> Kv(Rn) is upper semicontinuous if and only if there exists a sequence of continuous m-mappings {Fi}i= 1~ Fi: X ~ Kv(Rn), such that for every x @ X we have: 1) Fi+l(X) C Fi(x), i = 1, 2, . . . ; 2)F(x) C int Fi(x), i = 1, 2, . . . ; 3)F(x) - NFi(x). f=l 1,5.4. THEOREM. An m-mapping F: X --, Kv(R n) is lower semicontinuous if and only if there exists a sequence of continuous m-mappings {Fi}i=l ~, Fi: X ~ Kv(Ra), such that for every x E X we have: 1) Fi(x) C Fi+t(x), i = t, 2 . . . . ; 2) dim Fi(x) = dim F(x), i = 1, 2, . . , ; 3) F(x) = LJFi(x). i=1 1.5.5. THEOREM. Let F: X ~ Kv(R n) be an upper semicontinuous m-mapping, and let G: X ~ Kv(R n) be a lower semicontinuous m-mapping. In addition, assume that for every x E X we have F (x) ~ G (x).

Then there exists a continuous m-mapping H: X ~ Kv(Rn) such that for every x E X we have

F(x) ~H(x)

cG

(x).

Theorem 1.5.3 is closely related to the results of DeBlasi [509]. It was shown in [514] that there exist single- and multivalued continuous approximations in the sense of the distance between graphs in the Hausdorff pseudometric. Tsalyuk [317] considered Lipschitz multivalued approximations of upper semicontinuous m-mappings. Approximations of m-mappings by upper and lower semicontinuous step functions and m-mappings are considered by Spakowski in [1154] (see also [72]). Various types of single-valued and multivalued approximations were also studied in [326, 327, 404, 617].

w

Differentiation of Muitivalued Mappings One of Lhe most natural approaches to the concept of differentiation for m-mappings was proposed by DeBlasi.

861

1.6.1. Definition. An m-mapping F: X --, G(Y) of normed spaces is said to be DeBlasi differentiable at a point xo E X if there exists an upper semicontinuous m-mapping Gxo: X --, Cv(Y) such that: 1) Gxo(Xx) = XGxo(X), Vx E X, X ___ 0; 2) h(F(xo + ~x); F(xo) + Gxo( X)) = o( IIzXxII), where h is the generalized Hausdorff in C(Y). The m-mapping Gxo is called the DeBlasi derivative of the m-mapping F at the point xo. Further development of this idea is the subject of [434, 1099]. In particular, Spakowski [1153] proposed a scheme for constructing the derivative for m-mappings with values in a topological cone that naturally includes the DeBlasi derivative. Another abstract scheme for construction of derivatives (for mappings of quasilinear spaces) was proposed in [ 15]. Banks and Jacobs introduced the notion of a--differentiability (see [45]). Le Van Hot [818] extended this notion to m-mappings with values in locally convex spaces, and investigated differentiable sections of 7r-differentiable mappings. In [1034] the construct of ~--differentiability was extended to the case of fuzzy mappings. Guzzardi [652, 653] introduced the notion of differentiability of an m-mapping with respect to a cone: Let E be a real Banach space with a cone P. 1.6.2. Definition. A homogeneous upper semicontinuous m-mapping Gxo: E --- Kv(E) is called the upper Hdifferential of the m-mapping F: P -, Kv(P) at the point xo E P with respect to P if there exists a 6 > 0 such that F (x0 + lz) c F (x0) + Ox0 (h) + R (x0, h), where R(xo, -) is an m-mapping of P into E such that IIR(xo, h)[I = o( IIh II) for h -- 0. These papers apply this concept to proving the existence of positive fixed points for single- and multivalued map: pings, investigation of the topological structures of sets of solutions for operator equations, and to bifurcations of solutions. The directional derivative of m-mappings is considered in [156, 147, 221,236]. The effectiveness of this definition is illustrated on the example of a response problem for differential inclusions. Other definitions of differentiability of m-mappings and related problems are considered in [39, 40, 98, 121, 238, 242, 243,577, 622, 623, 727, 874, 875, 833, 1049, 1175]. for

IIh II <

w Measurable Muitivalued Mappings. Measurable Sections. Multivalued Integrals Let (T, ~) be a measurable space, and let Y be a metric space. 1.7.1. Definition. An m-mapping F: T --> C(Y) is said to be measurable if F -~ (V) = { t t T : F (t) NV=/= ~ } t E

for any open V C Y. Measurable m-mappings are widely used in convex analysis, the calculus of variations, optimal control theory, theory of differential inclusions, and other fields of modern mathematics (see the bibliography of [45]). The fundamental properties of measurable mappings were presented, for example, in [45, 46, 47, 235, 628] and elsewhere. Further development of research on properties of measurable m-mappings is the subject of a whole series of papers. Panteleev [215] proved an analog Luzin's C-property for measurable m-mappings (see, for example, [45, 47]) without assumptions on the compactness of the domain of definition. Ioffe proved [724] the following theorem on representation of measurable m-mappings. 1.7.2. THEOREM. Let (T, E, /z) be a space with a complete cr-finite measure, Let X be an uncountable Polish space, and let F: T --, C(X) be an m-mapping with uncountable images such that the graph r F belongs to E x Y3X , where Y~X/denotes the a-algebra of Borel subsets of X. Then there exist a Polish space Z and a mapping f: T x Z --, X that satisfy the Caratheodory conditions and are such that f(t, "): Z --, X is injective and f(t, Z) = F(t) for all t E T. A number of properties of measurable mappings that are useful for analysis of stochastic optimization problems were considered in [362]. Kusraev [145] proved the principle of openness for convex measurable m-mappings, which generalizes the known theorem of Schwartz on continuity of a linear operator with a measurable graph. Himmelberg's results on the properties of measurable m-mappings are extended and refined in [702].

862

In [656] there is a discussion of m-mappings of the form F(~o) = f(co) + r(w)B, where 9 is probability space, Y is a separable Banach space, f: 9 --, Y, r: 9 --, R+, and B is the closed unit ball in Y. It was shown that measurability of F is equivalent to measurability of f and r. Conditions for continuity of measurable m-mappings F of vector spaces that satisfy the condition F(l/2x + l/:y) D lhF(x) + ~AF(y) for all x and y were studied in [1177]. Salinetti and Wets [1093] presented a number of criteria for convergence of sequences of measurable m-mappings almost everywhere. They present an Egorov-type theorem for convergence of m-mappings with respect to a probability measure.

Maritz [859] found the following solution to a problem stated by Kuratowski: Let each closed subset of a topological space X be a G~-set, and let Y be a separable metric space. If the m-mapping F: X --, K(Y) is upper or lower scmictmtinuous, it is an m-mapping of Baire class 1 (i.e., the preimages of open sets are sets of type Fo). Completeness of spaces of semimeasurable m-mappings were considered by Lipovan in [837]. Various properties of measurable m-mappings were considered in [256, 365, 449, 807, 808, 828, 831, 832, 833]. 1.7.3. Definition. We say that a single-valued mapping f: T ~ Y is a measurable section of an m-mapping F: T P(Y) if f is measurable and f(t) E F(t) for all t E T. The existence of measurable sections is closely related to measurability of m-mappings. In particular, we have the following theorem of Castaing: 1.7.4. T H E O R E M . If (T, Z, /z) is a space with a nonnegative a-finite and complete measure and Y is a complete separable metric space, then measurability of F: T --, C(Y) is equivalent to the existence of a Castaing representation, i.e., a countable family {fn}n= 1~ of measurable sections of F such that U fn(t) is dense in F(t) for all t E T. n=l

Let 12 be a probability space, let Y be a complete separable metric space, and let F: 9 --, C(Y) be a measurable mmapping. In [363] there is a construction of two metrics in the space of measurable mappings from 9 into Y such that a sequence of sections that is dense in the family of all measurable sections of F with respect to the corresponding metric is a Castaing representation of F. A characteristic for convergence of sequences of m-mappings is given in terms of convergence of their Castaing representations. Problems on approximation of measurable sections of m-mappings were considered by Ricceri [1056]. In particular, the following assertion was proved: 1.7.5. T H E O R E M . Let T be a space with a measure, let (Y, p) be a separable metric space, and let F, F1, F 2. . . . be a sequence of measurable m-mappings from T into Y with complete images. The following conditions are equivalent: 1) for every measurable section f of the m-mapping F, there exists a sequence (fn is a measurable section of Fn~ that is pointwise convergent to f, and 2) for any x E X, y E F(x): limn__,oop(y, Fn(x)) = 0. The relationship between convergence of m-mappings and convergence of their measurable sections was also described in [1093]. Convergence of measurable sections in connection with problems on stochastic optimization was studied in [1071]. Papageorgiou [978] investigated problems on stability of sets of integrable sections of m-mappings. In [460] Castaing obtained the following generalization of his results to the inseparable case: 1) there exists a /~measurable section of an m-mapping F of a compactum T with positive Radon measure/z that has nonempty weakly compact images in a Banach space and is weakly #-measurable; 2) the set of/z-measurable sections of a weakly /z-measurable mmapping F defined on T and taking nonempty convex weakly compact values in a weakly compact subset of a Banach space is a convex weakly compact subset of LEI(T,/z). Let T be a space with a measure, let X be a Polish space, and let Y be a separable Banach space; an m-mapping F: T • X --, Cv(Y) is weakly measurable lower semicontinuous in its second argument. Rybinskii [1076] proved the existence of a measurable section of F that is continuous in the second argument. In addition, it was also shown that such sections form a Castaing representation for F. The existence of Caratheodory-type sections for m-mappings with convex closed images that are measurable in the first argument and continuous in the second was considered in [807, 808]. In [683] Hansell considered a generalization of the Kuratowski-Ryll-Nardzewski theorem on the existence of measurable sections, and in [684], studied its equivalence to the principle of countable reduction. Analogs of the Kuratowski-Ryll-Nardzewski theorem were also obtained by Idzik [720].

863

The monograph [1155] investigates the existence problem for measurable sections from the viewpoint of modern descriptive set theory. It presents a theorem generalizing the Kuratowski-Ryll-Nardzewski result and presents necessary and sufficient conditions to be satisfied by the graph of an m-mapping for existence of measurable sections. A theorem on conditions for existence of measurable sections of m-mappings inverse to continuous mappings of compact spaces was proved by Lotz [843]. Himmelberg, Van Vleck, and Prikry [704] obtained a number of results on existence of measurable sections for mmappings in complete metric spaces. One of the features of this paper is a systematic discussion of a class of mappings that is narrower than measurable, namely, mappings that are representable as the limit almost everywhere of sequences of mappings with a finite number of values, each of which is in a measurable space, and this constraint is imposed on both the m-mappings and their sections. Sosulski [1148] proved the following theorem on measurable sections: 1.7.6. THEOREM. Let (Y, p) be a separable metric space, T = [0, a]. Assume that the m-mapping F: T --, K(Y) is continuous and the mapping w: T --, Y is measurable. Then there exists a measurable section f of the m-mapping F such that

p(w (0, f (t)) =p(w(0, ~(t)) for almost all t E T. We should note the survey [640] by Graf on the theory of measurable sections of m-mappings. Together with a discussion of classical problems (theorems of Kuratowski and Ryll-Nardzewski, Castaing, Filippov, etc.), it considers new results on measurable sections of m-mappings of whose ranges are neither separable nor metrizable. Development of Filippov's implicit function theorem (see, for example, [45, 47]) is the subject of [184, 234]. Various conditions for existence measurable sections are also considered in [163,383,452,554,569,598,606, 608, 683, 684, 786, 828, 1006, 1136, 1156]. The existence of Borel sections for upper semicontinuous m-mappings was considered in papers by Jayne and Rogers [739-741]. They showed, in particular, that upper semicontinuous m-mappings of a metric space into a Banach space whose images are nonempty subsets of weakly compact sets have Borel sections. Borel sections of m-mappings were also investigated in [ 114, 464-466, 641,848, 854, 855, 1135, 1137, l 152]. We should also note the surveys [451] and [826]. Various properties of measurable sections have been described in a number of papers. Let T be a space with a measure, let Y be a separable Banach space and assume that F: T ~ Kv(Y) is measurable. Phan Van Chuong [1002] considered analogs of an infinite-dimensional form Lyapunov's theorem on density in sets of measurable sections of m-mappings F of sets of measurable sections of m-mappings F: /7(r

y is an extremum point of F ( ~ ) } .

Olech [954] noted that the set of all measurable sections of a measurable m-mapping is decomposable. He also investigated the properties of such sets. The relationship between compactness of a set of summable sections of an m-mapping and compactness of its images was studied by Assani and Klei (see [366]). Criteria for weak compactness of sets of measurable sections of m-mappings were given in [976]. Ceder [463] characterizes convex-valued m-mappings on the real line that admit sections mapping any interval in an interval (Darboux intervals). Beer [405, 407] investigated the existence of measurable single-valued approximations of m-mappings (in the sense of closeness of graphs in the Hausdorff metric), Doberkat (see [550]) used measurable sections in abstract automaton theory. The problem of control by a section was considered by Kuliev [137]. Multivalued measures and their properties were studied in [6, 274, 718, 956, 972, 1027, 1035]. 1.7.7. Definition. Let F be an m-mapping of space with a positive measure (T, E, /z) into a Banach space Y. The integral of F with respect to the measure/z is the set of all integrals of summable sections of F, i.e.,

T

864

The properties of multivalued integrals have been investigated in numerous papers. We should note the detailed discussion of the theory of integration for m-mappings in the monographs [47, 236, 628]. In particular, [236] presents several approaches to defining multivalued integrals and compares them. Polovinkin [237] studied the properties of Lebesgue integrals and obtained a generalization of Lyapunov's theorem on vector measures for m-mappings. He obtained necessary and sufficient conditions for existence of Riemann integrals and described this integral for nonconvex-valued mappings. He gave sufficient conditions for Riemann integrability of crosssections and geometric differences of m-mappings. In [736] there is a new definition of the Riemann integral for m-mappings, and its fundamental properties and connection with previous definitions are considered. Silin [265] found short and simple proofs for Lyapunov's theorems on convexity and compactness of integrals for m-mappings and insertion of subdifferentials in integrands. A new proof of the fundamental property of multivalued integrals that includes proving its coincidence with the convex hull of a mapping is given by Clarke in [482]. The same property in nonreflexive Banach spaces was studied by All Kahn [346]. In [460] Castaing proved convexity and compactness for the integral of a weakly upper semicontinuous m-mapping defined on metrizable compactum with positive Radon measure and having nonempty convex compact images in a Banach space. A theorem on convexity of the closure of integrals of m-mappings with ranges in separable reflexive Banach spaces, a theorem on representation of multivalued integrals, and an analog of the Lebesgue theorem on convergence of integrals of sequences of m-mappings are considered in [1033]. An analog of the Gould integral is constructed in [501] for m-mappings into a Banach space with closed convex bounded images. Certain properties of the Lebesgue integral and criteria for Riemann-Stieltjes integrability are considered in [20]. Gel'man and Gliklikh (see [73]) constructed a multivalued analog of the Ito stochastic integral and studied its properties. This integral was used to investigate stochastic differential inclusions. The integral of interval-valued functions was defined by Moore in [892]. The properties of multivalued integrals with variable upper limits were considered by Papageorgiou [974]. Application of multivalued integrals in optimal control problems is considered in [555, 556]. Application of quadrature formulas for approximate evaluation of Riemann integrals of m-mappings is considered in [19]. The theory of integration for multivalued additive set functions is studied in [217]. Lipovan [835] constructed an integral for multivalued functions by using the concept of probabilistic submeasure. One of the objects of nonsmooth analysis was investigated by Ioffe in [977], where he introduced the term "fan." An m-mapping of vector spaces F: X --, Pv(Y) is called a "fan" if 0 @ F(0), and for all X > 0 and x E X, we have F(Xx) = hF(x) and F(x 1 + x2) C F(xt) + F(x 2) for xl, x2 E X. This paper studies m-mappings G(t, x) that are measurable with 9 respect to t and fans with respect to x. He investigated properties of the integral

~) (~) = i O (~o, x) d r (co). s

Various properties of multivalued integrals were also considered in [202, 783,830, 860, 999, 1125].

CHAPTER 2 APPLICATIONS OF THE THEORY OF MULTIVALUED MAPPINGS Recent years have seen energetic application of multivalued mappings in traditional fields (game theory, mathematical economics, differential inclusions) and more modern areas (optimal control theory, variational inequalities, monotone operators, integral inclusions). We will review some of this research below.

w

Applications in General Topology

Multivalued mappings provide a convenient tool for investigating the properties of various topological spaces and their transformations. Fedorchuk [298], Dranishnikov [89, 90], Nepomnyashchii [182, 921], Koyami [790], and Suzitskii [1161] have described properties of multivalued absolute retractors and extensors, i.e., properties of spaces that admit multivalued retractions or continuations of specific classes. Problems on continuous continuation o f m-mappings were studied in [487]. 865

Lisitz [839], Koyami [790] and Suzitskii [1161] presented applications of semicontinuous multivalued mappings in shape theory. Obukhovskii and Skaletskii [197] investigated problems on continuability of mappings from subsets of products of topological spaces that are the graphs of upper semicontinuous m-mappings. We should note the following fundamental results. Let X and Y be topological spaces; the subset A C X x Y belongs to the class 9~(X x Y)[9~c(X x Y)] if A is the graph of some upper semicontinuous (continuous) m-mapping F: X -, K(Y). 2.1.1. T H E O R E M . Let X be a T-collectively normal and countably paracompact space, and let Y be a metric space. Also, let E be a Banach space of weight _< T. If "A E jdE (X • Y), then any continuous mapping f: A --, E has a continuous extension {': X • Y --, E. 2.1.2. T H E O R E M . Let X be a ~--collectively normal space, let T be a closed countably paracompact subspace of X, let Y be a metrizable space, and let E be a Banach space of weight <_ r. If A E 3r • Y) and A ('1 ((X~T) x Y) E r c((X~T) • Y), then an arbitrary continuous mapping f: A -~ E can be continued to X • Y. We should note that this theorem is also true when X is a completely regular space. We need only require that the subset T be bicompact. 2.1.3. T H E O R E M . If X is a normal space, Y is a metrizable space, and A E Jd(X x Y), then an arbitrary function f: A --, R that is continuously uniform on Y can be extended to X • Y. This implies, in particular, the following generalization of Starbird's theorem on continuation of homotopy. 2.1.4. T H E O R E M . Let X be a normal space, let Y be a metrizable space, and let T C X be closed. If A E 9~(X X Y), A (3 (T X Y) E 3g'c(T x Y) and A f3 ((X\T) x Y) E ~ c((X\T) • Y), then f: A --, R can be continued to X x Y. Applications of m-mappings on continuing continuous mappings were also considered in [827]. The properties of openness and completeness of continuous mappings f of topological spaces were investigated in [844] by means of m-mappings inverse to f and mappings induced by f on spaces of subsets. The problem of preservation of paracompactness under certain classes of m-mappings was studied by Kovachevich in [787]. Joseph (see [744]) characterized regularity and normalness of topological spaces in terms of m-mappings. A new criterion for bicompactness of topological spaces in terms of upper semicontinuous m-mappings was obtained in [572]. M-mappings corresponding to topological closure operators on sets were characterized by Kaminskii [747]. A set in a Hausdorff topological space is said to be K-analytic in X if it is the image of a Baire space of weight k under an upper semicontinuous m-mapping with compact images. A theory K-analytic set that is a generalization of the classical theory of analytic sets is constructed in [687]. Trokhimchuk, Zelinskii, and Sharko [293] considered problems associated with the study of m-mappings of topological and homological manifolds, as well as application of their results to the theory of holomorphic mappings.

w

Metric Projections

2.2.1. Definition. Let E be a normed space, let M be a closed subset of E, ~M: E -~ R+ ; ~M(X) = inf {]t Y -- x [l: y E M} is a distance function. The m-mapping PM: X --> P(M) p ~ ( x ) : {vcJVl : Llv-xll = ~ . ~ ( x ) } is called a metric projection (onto the set M). The basic types of continuity for metric projections are considered in [209, 210], which presents criteria for upper semicontinuity of metric projections into Banach spaces. Conditions for continuity of the metric projection in terms of differentiability of the corresponding distance functions are given in [593, 594]. Balagansldi [21] considered conditions for sequential weak upper semicontinuity of metric projections onto bounded sets in Banach spaces. Various conditions for upper semicontinuity and continuity of metric projections are also considered in [340, 962, 1230]. Continuous selections of metric projections were studied in [17, 423-425, 1147].

866

A survey on conditions for linearity and continuity of selections of metric projections may be found in [526]. In [948] there is a survey of results pertaining to the existence and uniqueness of continuous and Lipschitzian selections of metric projections onto finite-dimensional subspaces of C[a, b]. A theory of a certain class of m-mappings containing metric projections for which the fundamental metric properties of metric projections hold is constructed in [632]. Suppose E is a normed space, M E Kv(E), and g: M ~ M is a continuous mapping. Developing the notion of metric projection, Prolla [1028] considered the existence of points x E M such that g(y) is the best M-approximation of f(y) for a continuous mapping f: M ---- E, i.e., I1g(Y) - f(Y)11 = ~M(f(Y)). Because each such point is a fixed point of the m-mapping Af: M ~ P(M),

As(v) = [z~M : Ilg(z)-f(v)II~<m(v)}, where

m(v ) =+[llg(V)--[(J)I]-~q~z(f(b'))],

their existence is a consequence of the Bohnenblust-Karlinfixed-point

theorem. Let X be a normed space, and suppose that for H C X and t _> 0 the metric projection Prlt: X --, P(H) is given by the formula

p~(x)={VEtt:]lx--vjj..
H)}.

The properties of continuity and differentiability of the m-mappings PHt were studied by Berdyshev in [28-31]. Conditions are given in [1043] for upper and lower semicontinuity of m-mappings that associate, with a point in a normed space; points in the fixed subset furthest from it. Various properties of metric projections were also studied in [431,631, 639, 962, 1118-1120, 1144, 1222, 1230]. w

Results from the Theory of Differential Inclusions By a differential inclusion we mean a relation of the form

x' (t) ~ (& x (t)),

(1)

where cI, is an m-mapping. Solutions of differential inclusions are classified as follows: an absolutely continuous function x(.) is called a Caratheodory solution of (1) on some segment I if (1) is satisfied almost everywhere on I. By a proper solution we mean a solution whose derivative x'(.) is Riemann integrable. A classical solution is a continuously differentiable function x(') such that inclusion (1) is satisfied for all t E I. The last ten years have seen rapid development of the theory of differential inclusions, which has found broad application in many areas of contemporary mathematics. We have already noted its application in the theory of optimal control in See. 2.4. Differential inclusions are considered in the study of differential equations with discontinuous right sides [296, 299, 627, 1168], mathematical economics [1206], differential games [134], polling theory [2], in planning dynamic population development [1209], and in other problems [70, 868, 965]. We should note the following important recent publications on the theory of differential inclusions and its applications: V. I. Blagodatskii and A. F. Filippov [33], A. A. Tolstonogov [289], Aubin and Cellina [370], Glodde and Niepage [628]. Various aspects of this theory have also been considered in survey articles and monographs [42, 46, 47, 100, 236, 299, 371,455, 703]. Existence Theorems for Solutions. A large number of existence theorems have been proved for differential inclusions subject to various assumptions on the relationship between the right side and (t, x) and the structure of its images. We will present several rather typical and general examples that are not too cumbersome, limiting our discussion to Caratheodory solutions and local existence theorems. We first consider inclusions in finite-dimensional spaces. Let I = [to, to + a]; B = (x E Rn: Ilx- xoll < b} and consider the Cauchy problem

x' (t) ~F (t, x (t)),

(2)

x(to) =x0.

(3)

867

We have the following propositions: 2.3.1. THEOREM (Davy, Ref. Zh. Mat., 12B195, 1972). Assume that the m-mapping F: I • B --- Kv(Rn) is such that: a) for almost all t E I, the m-mapping F(t, -): B ~ Kv(R n) is upper semicontinuous; b) for each x E B, the m-mapping F(', x): I --, Kv(Rn) has a measurable section f, and there exists a summable function IIe(t)II <-- m(t) for almost all t E I and to+,,

f m(t)dt < b. Then problem (2)-(3) has a Caratheodory solution in the segment [to, to + c~]. to 2.3.2. THEOREM (Olech, Ref. Zh. Mat., 8B217, 1976). Assume thatthe m-mapping F: I • B --- K(Rn) is such that: a) for almost all t E I the m-mapping F(t, .): B --, K(R n) is lower semicontinuous, and at those points x E B where the set F(t, x) is not convex, it is continuous; b) for each x E B, the m-mapping F(., x): I ---,K(Rn) is measurable; c) there exists a sup summable function m such that IIF(t, x)II =yEe(,,*i

"IIYII -< m(t), v x E B and f m(t)dt < b. Then problem (2)-(3) has to

a Caratheodory solution on the segment [Io, to + ed. 2.3.3. THEOREM (Bressan [439], Fiacca [585]). Assume that the m-mapping F: I • B --, K(Rn) is lower semiconto+tt

tinuous and there exists a summable function m such that IIF(t, x)[I _< re(t), vx E B and

f m(t)dt < b. Then problem (2)-

to (3) has a Caratheodory solution on the segment [to, to + ~]. As a rule, finite-dimensional existence theorems have analogs for differential inclusions in infinite-dimensional spaces. As an example, we will present the infinite-dimensional form of Theorem 2.3.1 (see [289], Theorem 2.5.4). Let E be an infinite-dimensional Banach space, let X be the Hausdorff noncompactness measure in E; I = [%, to + a]; and let B = {x E E: IIx - xoll < b}. Let o~: I x R+ --, R+ be a Kamke function, i.e., 60 is a Caratheodory function that is integrally bounded on bounded subsets of [c, a] x R+, c > to, such that: a) ~0(t, 0) = 0 for almost all t; b) for each c, % < c < a, the only absolutely continuous function r: [to, c] --, R+, r(to) = 0, that satisfies the differential equation r'(t) = oa(t, r(t)) almost everywhere on [to, c] is r(t) -= 0. 2.3.4. THEOREM. Let the m-mapping F: I • B ---,Kv(E) be such that: a) for almostall t E I, them-mappingF(t, .): B ~ Kv(E) is upper semicontinuous; b) for each x E B, the m-mapping F(., x): I + Kv(E) has a strongly measurable cross section; c) there exists a summable function m such that IIF(t, x)II -< m(0 vx ~ B; d) for almost all t E I, we have x(F(t, D)) < oa(t, x(D)) for any D C B. Then there exists a Caratheodory solution of problem (2)-(3) on some segment [to, to + ~]. For problems concerning solvability of the Cauchy problem for differential inclusions in Banach spaces, the properties of solutions and the relations between sets of solutions, etc., see Tolstonogov's monograph [289] and [277-288]. Existence theorems for solutions of the Cauchy problem for differential inclusions are considered in [11, 81, 84, 148, 149, 172, 177, 241,290, 292, 305, 312, 313,328, 330, 359, 360, 369, 398,440, 457,585,605,777, 824, 841,857, 861, 9801. Existence theorems for differential-functional inclusions may be found in [51, 52, 54, 57, 86, 105, 108, 203, 291, 306-308, 530, 612, 621,746, 772, 774-776, 779, 845, 939, 955, 1186, 1188, 1210]. Existence theorems and the properties of solutions for differential inclusions are also considered in [7, 10, 300, 302, 305, 489, 491,492, 496, 814, 815, 947, 1005, 1109]. Differential equations and inclusions that cannot be solved for derivatives were the subject of [309, 310, 941,942, 1186]. Differential inclusions in partial derivatives were studied in [64, 66, 402, 471, 759, 873, 877, 1149-1151, 1170, 1171]. Properties of Solutions and Integral Funnels of Differential Inclusions. The compactness and connectivity of sets of solutions for differential inclusions, their dependence on the initial conditions and the right sides of the inclusions, and other issues in the qualitative theory of differential inclusions have been studied by many authors: [35, 62, 106, 109, 205, 206, 208, 214, 224, 240, 269, 273, 278, 280, 283, 284, 289, 331,429, 467, 512, 609, 636, 659, 778, 802, 812, 1085]. The connection between solutions of inclusions with nonconvex right sides and solutions of equations with convexified right sides was considered in [281,282, 287, 289, 290, 329,418, 516, 771,780, 832].

868

The extremal properties and other properties of solutions of differential inclusions were investigated in [10, 141,204, 288, 289, 458, 643,657, 658, 842, 953]. The equation of the integral funnel of a differential inclusion was studied in [59, 285, 286, 289]. The problem of constructing a differential inclusion from a set of its solutions was considered in [303, 735, 812]. Other Problems in the Theory of Differential Inclusions. The averaging problem for differential inclusions was studied in [112, 226-229]. We will present a result of V. A. Plotnikov (see Ref. Zh. Mat., 2B226, 1980). Consider the differential inclusion

xEeF(t, x), x(O) =x0, where F(t, x) E K(R n) for all (t, x), and t is a small parameter. Let

y ~F(v), y(O)=x0 $

be an averaged differential inclusion, where 15(y) =

l~(1/s)fF(t,.|y)dt (the limit is taken in the sense of the Hausdorff 0

metric).

2.3.5. THEOREM. Assume that the m-mapping F is continuous with respect to (t, x), Lipschitzian with respect to x, and uniformly bounded. Assume that the limit F(y) exists at each point y C R n. Then, for any ~ > 0 and l > 0, there exists an Co(~, /) > 0 such that for all 0 < e _< co, the following assertion is valid: For any solution x(0 of the initial differential inclusion, there exists a solution y(t) of the averaged differential inclusion such that

IIx(t)-v(t)Ilk
w Pontryagin's Principle of the Maximum for Differential Inclusions and Optimization Problems The theory of differential inclusions is closely related to optimal control problems, so it is completely natural for many authors to attempt to apply Pontryagin's principle of the maximum to the case of differential inclusions. As an example, we will present an optimization problem for a differential inclusion with an integral quality criterion. Assume that we are given: a differential inclusion x' E F(t, x), a set of initial states Mo C R n, a set of final sets M 1 tl C R n, and an integral quality criterion ~(x(.)) =

fro(t, x(t))dt,

where fo: RI x R n ~ R n is some known function. It is

to

869

required to find a solution x(t) of the differential inclusion in the interval [to, tl] that will assure transition from the set of initial states Mo to the set of final states M 1 and minimize the quality criterion. The first existence theorem for such an optimal solution for a differential inclusion was proved by A. F. Filippov. We will present this theorem, following [33]. Assume that the phase constraint x(t) E X for any t E [to; tl] is imposed on the solution x(t) of the differential inclusion. 2.4.1. THEOREM. Assume that the m-mapping F: R 1 x R ~ --, Kv(R n) is measurable with respect to t, upper semicontinuous with respect to x, and satisfies the bound

c(F(t, x);

x) ~/e (t) (1-Fllxll2),

where c(F(t, x); -) is the support function of the set F(t, x). Assume that the sets Mo, M1, and X are closed, and that at least one of the sets M o or M 1 is bounded, and assume that the function fo(t, x) is measurable with respect to t and continuous in x. If there is at least one solution x(t) to the differential inclusion that satisfies the conditions x(to) E Mo, x(h ) E M1, x(t) E X, then there exists an optimal feasible solution that minimizes the quality criterion o~. Blagodatskii [32] considered necessary and sufficient conditions for optimality. This paper permits us to state the following result. We augment the phase coordinates x 1. . . . , Xn, which vary according to the rule x'(t) @ F(t, x), by an additional coordinate xo that satisfies the equation xo' =fo(t, x). The vector x = ( x 0 , xl . . . . .

x ~ ) = ( x 0 , x)ER n~

satisfies the inclusion .~' E F(t, ~), where F(t; .~) = {(Yo; Y): Yo = fo(t, x), y E F(t, x)}. Consider the set lq o = {(Yo, Y): Yo =0; y E Mo} , lVl1 = {(Yo, Y): Y E M1, Yo E R1}. Then the optimization problem with an integral quality criterion can be stated in the following equivalent form: Among all solutions .~(t) = (Xo(t), x(t)) of the inclusion 2' E l~(t, ~) that assure transition from the set lflo to the set lVl1 in the interval [to, tl] , find one for which the coordinate YCo(tl)is minimal. 2.4.2. Theorem (Principle of the Maximum). Let ~(t) = (xo(t), x(t)) be an optimal solution for a differential inclusion. We assume that the point t I is a proper point of the function x'(t), that the set Mo is locally convex at the point x(to) E Mo, and that the set M 1 is locally convex at the point x(tl) E M 1. Then there exists a nontrivial solution if(t) = (fro(t), if(t)) of the conjugate differential equation

~'(t)~-

ac(-P(t,~(t));-~(t)) ax

that satisfies the following conditions: 1) maximum condition: (x' ([), 3 ( [ ) ) = c (F(t, .~(~)); ~ (l)) for almostall

felt0, t11;

2) transversality condition on Mo:

c(T(Mo, X(to)) ; r

(re)) =0,

where T(Mo, x(to)) is the space tangent to Mo at the point X(to); 3) transversality condition on the set MI: c (T (MI, x(t~) ) ; - - , (t,)) = 0, where T(M1, X(tl)) is the space tangent to M 1 at the point x(tl). In addition, the function c(F(t, 2(t)); ~(t)) is equal to zero when t = t 1, and the function ~o(t) is constant and nonpositive. Various aspects of the theory of optimization for differential inclusions were also included in [36, 37, 74, 78, 83, 111, 124, 136, 139, 166, 168, 207, 236, 239, 244, 245,249, 250, 257,258, 266, 271,289, 372, 415, 422, 438,470, 485, 486, 583, 599, 600, 603, 698, 753, 862, 981, 1196]. Other applications of muitivalued mappings in optimization problems were considered in [74, 135, 165, 171, 252, 253, 495, 614, 710, 754, 809, 1066, 1067, 1069, 1167].

870

w

Integral Inclusions

Let U C R n be bounded and convex, and let F: [0; h] • 0 --, R n be continuous, continuously differentiable with respect to the second variable, and such that det fu'(t, u) ;~ 0 u E [0; h]. Assume that the m-mapping F: [0; h] • 0 --, Kv(Rn) satisfies the Caratheodory conditions. Benkafadar [24] used the concept of the local degree of a multivalued vector field with Fredholm principal part to study solvability of inclusions of the form

f (t, ~t(t))~g

Re (it) (s) ds ,

where g: C[0, h] ~ C[0, h] is a continuous mapping and PF is the multivalued superposition operator generated by F. T. A. Sventsitskii [259] studied Volterra-type integral inclusions t

x(t)~xo+ I F ( t , s, x(s))ds,

t~i0;T],

0

where x: [0; T] --, R n is the desired function, and F: [0; T] • [0; T] • R n ---, Kv(R n) is some m-mapping that is continuous in its first argument, measurable in its second, and upper semicontinuous in its third. Conditions were given for convergence of a method of successive approximations and a theorem was proved on the continuous dependence of the funnel of solutions on the initial value xo and the m-mapping F. These results were extended in [260] to the case of integral inclusions with lagging arguments. Problems on local solvability, extension of solutions, and the connectivity of sets of solutions of an integral inclusion were studied in [55-57]. The existence of solutions for integral inclusions with pseudomonotonic operators in Hilbert space was proved by Hirano (see [708]). Random integral inclusions in a Banach space were studied by Phan [1003, 1004]. Dombrowska [505, 506] studied integral equations of the form

[o(0, [eI-~; o1

x(t)= [o(o)+ !

X)as,

tG[o;rl,

where the m-mappings ~, F, and X operate in Kv(R n) and are continuous in the Hausdorff metric. He investigated the existence of solutions and their properties. The theory of inversion of multivalued vector fields was used by Povolotskii and Sventsitskii [232, 233] to investigate the characteristic values and vectors of multivalued integral operators of Hammerstein type that are almost linear. Kostoyusov [130] constructed an integral inclusion describing the limiting trajectories of sliding-pulse modes. A fixed-point theorem for m-mappings was used to investigate the existence of solutions for systems of integral equations with discontinuous right sides in [63, 67, 68]. Various existence theorems and properties for integral inclusions were also considered in [247, 261,342,352, 644].

w

Applications to the Theory of Differential Equations

The convergence of problems in the theory of ordinary differential equations to differential inclusions can be illustrated by the fact that any implicit differential equation

f(t, x, x')=0 can be interpreted naturally as a differential inclusion,

x'~F(t, x) ={y : f(t, x, y) =0}. Problems of this kind were considered by I. A. Finogenko [310]. Investigation of differential equations by means of differential inclusions was also the subject of [304, 1096].

871

Differential inclusions were used in [379] to study the stability of solutions to differential equations. Kiffe [766] used m-mappings to investigate solvability of integral Volterra equations. The mixed problem for semilinear hyperbolic systems with multivalued boundary conditions was considered in [396, 897, 898]. This problem reduces to inclusion with a maximal monotonic operator in a Banach space. Chang [473] used the theory of the topological degree of m-mappings to investigate such typical problems with free boundaries as problems with obstacles, the problem of water leaking from a surface, Stefan's problem, and the Elenbaas equation. A mathematical model for thermal propagation with controlled temperature was considered in [625,626]. This model reduces to a boundary-value problem for the equations of thermal conductivity, which, in turn, leads to an integral inclusion. The classical Bohnenblust-Karlin theorem is used to prove the existence of a solution. Various applications of m-mappings in the theory of differential equations were also considered in [65, 85,437, 462, 498, 578, 1058, 1112].

w

Monotonic and Accretive Operators Let E be a Banach space, and let E* be its adjoint. We consider the bilinear form on E* • E defined by the condition <w; u > = w ( u )

for any w E E*, u E E. 2.7.1. Definition. A set G C E* • E is monotonic if, for any pairs (wl; ul) and (w2; u2) belonging to G, we have (w2 -- wfi u 2 -- ul) -~ 0. A monotonic set that is not a subset of a larger monotonic set is said to be maximally monotonic. 2.7.2. Definition. An m-mapping F: Y C E --, P(E*) is said to be monotonic if its graph F F C E x E* is a monotonic set. If, moreover, F F is maximally monotonic, then the m-mapping F is said to be maximally monotonic. Maximally monotonic operators were studied in [796, 799]. An important example of a monotonic mapping is the dual mapping. 2.7.3. Definition. A mapping J: E --- P(E*) is called a dual if, with any vector x E E, we can associate

J(x)={w~E*:<w; x>=llx[I 2, {Iwll=ilxll) It is clear that for any x E E, the set J(x) is nonempty, convex, and closed. Since J(x) C S* = {u E E*: 11u [I =

IIxll~, the d~al is single valued when the space E* is strictly convex. Various properties of dual mappings were studied in [364, 426, 1030]. Another important example of multivalued monotonic mappings is the subdifferential of a convex function. Let f: E --, ( - ~ ; + ~ ] be a lower semicontinuous convex function that is not identically equal to + ~ . 2.7.4. Definition. A vector w E E* is called the subgradient to f at the point % E E if f(uo) is finite and f (u)--~(u0) ~><w; U--Up> for any u E E. The set of all subgradients to f at the point u o E E is called the subdifferential and denoted by 0f(uo). Subdifferentials are monotonic mappings. This notion has broad application in contemporary nonlinear analysis. Various properties of subdifferentials were studied in [22, 144, 150, 155, 192, 211, 430, 950, 969, 1064, 1070, 1174, 1205] (see also the appendix to this survey). In particular, Krauss [796] proved that every maximally monotonic operator is the subdifferential of a characteristic concave-convex function. 2.7.5. Definition. An m-mapping F: D(F) C E --, P(E) is said to be an accretive operator if for any xl, x2 E D(F), Yt E F(xl), Yz E F(x2), and r > 0, we have

I[Xl--X2l[~llxt--x2q-r(Yt--Y2) [[. If the range of the operator i -- rF coincides with all of E for any r > 0, then F is said to be an m-accretive operator. Various properties of accretive and monotonic operators were studied in [559, 563, 630, 768, 769, 793, 794, 798, 799, 800, 994, 1037, 1038, 1207, 1223]. 872

Operator Inclusions with Monotonic and Accretive accretive principal parts was studied in [752, 784, 895, 982, Kartsatos [752] proved a series of theorems on surjectivity of m-accretive and C: D(F) --> P(E) is compact. Morales [895] proved that a multivalued m-accretive continuous function ~,: R+ --, R+ such that 1) ~p(0) = 0, ~(r) > 0 for r > 0; 2) liminf~(r) > 0;

Operators. The range of certain operators with monotonic or 1184, I229], and later shown to be surjective. For example, mappings of the form F + C, where F: D(F) C E -, P(E) is mapping F: D(F) C E --, C(E) is surjective if there exists a

3) II u - v II >-- y U) for any x, y E D(F), u E F(x), v E F(y). This assertion is generalized to a broader class of spaces. Solvability of inclusions of the form

F (x) 9f, where F is an m-mapping with a monotonic or accretive principal additive part, is the subject of [472, 477, 613, 647, 648, 755, 797, 847, 894, 1048, 1063, 1072, 1146, 1183, 1185]. The properties of a sequence constructed with the proximal-point algorithm of Rockafellar (Ref. Zh. Mat. 5B570, 1977) are studied in [755], and it is proved that it converges to the solution of the inclusion O E F(x), where F is a maximally monotonic operator. The topological degree is constructed in [1228] for operators of the form F + P, where F is a maximally monotonic operator and P is a generalized pseudomonotonic operator, its properties are investigated, and applications to surjectivity for these operators are given. The topological degree of m-mappings of monotonic type is also considered in [441,795]. The article [1048] studies inclusions of the form 0 E fix) + 0Co(x), x E R n, and its perturbations 0 E f(x, p) + 0r , p), p E P C R s. An analog of Sard's theorem is proved for m-mappings of the form f(x, p) + 0r where P is a smooth manifold and f is a mapping of the class C 1. The asymptotic behavior of the solutions of second-order difference equations with maximally monotonic operators in Hilbert space is studied in [888]. Differential Inclusions with Monotonic and Accretive Operators. The study of differential inclusions with monotonic and accretive operators is the subject of a large number of publications. Solvability of the Cauchy problem for certain classes of differential inclusions containing accretive or monotonic operators is the subject of [311,469, 518, 568, 596, 650, 651,705, 1126, 1199-1203]. We will cite several results. For example, Cellina and Marchi [469] considered the Cauchy problem

x'~--Ax+F(t, x), x(a) =x0, where A: D(A) C R n --- P(R n) is a maximally monotonic operator, and F: [a; oo) • i)(A) --- K(R n) is an m-mapping that is continuous in the Hausdorff metric and admits the linear bound

tiE(t, x ) I I ~ ( 0 lixll+f~ (0. It is proved that under these assumptions, the Cauchy problem has a solution that is defined on [a; oo). V. L. Khatskevich [311] considered the problem

u'+F(t, u)~O, u(O)=uo, in a Hilbert space H, where f: [0; r Let

• H ~ Cv(H) is a multivalued mapping.

f (t, u ) = ve~" (t, ~),

IIv ii-- max iI'a' i!.

873

2.7.6. THEOREM. Assume that for each t E [0; o~], the operator F(t; -): H --, Cv(H) is maximally monotonic, and the mapping f(', x): [0; ~] --, H is continuous for any x E H. Assume that for every ball IIx II -< r, there exists a summable function Or(0 such that for almost all t E [0; r

Ilf(t,x) ll<~c~(t) (u

I[xll<~r),

Then the Cauchy problem has a unique solution for every uo E H. The publications [341,443,444, 454, 459, 481,483, 497, 517, 529, 531,570, 629, 706, 714, 729, 731,756, 762, 887, 952, 957, 958, 990, 1009, 1010, 1095] consider stability, asymptotic behavior of solutions, existence of periodic solutions, and other issues in the theory of differential inclusions with multivalued monotonic and accretive operators. Stochastic differential inclusions in Banach spaces containing maximally monotonic and accretive operators are considered in [131-133]. 'Various other problems on differential inclusions with monotonic and accretive operators were studied in [413, 571, 692, 791,901,988,989, 1098, 1202, 1203]. Other Problems. Various classes of equations and partial differential inclusions containing monotonic and accretive operators are considered in [99, 339-401, 419-421, 532, 533, 574, 589, 691, 693, 757, 758, 849, 899, 900, 959, 1090, 1121, 1166, 1181]. Integro-differential inclusions are considered in [525, 645, 719]. Variational inequalities with monotonic operators and the properties of their solutions are studied in [445, 707, 909, 913, 935]. Optimization problems for differential inclusions and variational inequalities with maximal monotonic operators are investigated in [393-395, 1179, 1180, 1182]. w

Variational Problems

Variational inequalities have interesting applications to problems in mechanics, thermal conductivity, diffusion theory, optimal control theory (see, for example, the monograph of Baiocci and Capelo in the bibliography of this appendix). Investigation of variational inequalities is closely related to multivalued mappings: As an example, we consider the following simple existence theorem (see [730, 1227]). 2.8.1. T H E O R E M . Let X be a topological vector space, M E Kv(X), and assume that f: M --> X* be continuous. Then there exists a uo E M such that f (Uo) (uo--u) =

C(u0), Uo--u>~o

for all u E M. This proposition is proved by assuming, to the contrary, that with each v E M we can associatea nonempty set F(v)

C M, F(v) = {u E M: (f(v), v -- u) < 0}. Each F(v) is convex and F-l(u) is open for any u E M. As a result, it follows from the Browder fixed-point theorem (see Theorem 2.1 in the first part of our survey, and the remark on it), F has a fixed point v o E M, v o E F(vo), i.e., (f(vo, v o -- v o) < 0, which is a contradiction. Application of techniques from the theory of multivalued mappings to solvability and investigation of variational inequalities is also the subject of [442, 447, 448,519, 564, 654, 728, 817, 1060, 1123, 1165]. The continuity properties of solutions to variational inequalities were studied in [934], and [707] contains an investigation of homotopic invariants of solvability for nonlinear variational inequalities. The problem of optimal control for systems described by variational inequalities has been investigated in [393-395]. In [728] the K-variational problem is investigated in a Banach space X as a pair (F, g) consisting of an m-mapping F with range in the dual space X' and functional g: G C X --, ( - o o , + ~ ] such that for each f E X', the inclusions u E D(f) and f E F(u) occur if and only if g(u) -- {f, ku) _< (gv) -- (f, kv) for any v E G, where k: D(k) C X --, X is some linear operator. An equivalent statement of Ekeland's variational principle (Ref. Zh. Mat., 2B781, 1975; 1B577, 1980) is given in [983] in terms of fixed points of m-mappings. The connection between Ekeland's principle and the fixed points of m-mappings is also studied in [1162, 1221]. Multivalued mappings are applied to the problem of optimizing functionals in [165, 230, 321,403,711, 1068].

874

w

Applications to Game Theory and Mathematical Economics

Historically, game theory was one of the first applications of multivalued mappings. In fact, the fundamental theorem of game theory -- the minimax theorem -- is a direct consequence of fixed-point theorems for m-mappings (see [46, 47, 564, 1227] and others). The application of the theory of multivalued mappings to derivation of various minimax relations is used in [442, 447, 448, 519, 642, 654, 670, 673, 681,712, 713, 785, 817, 1059, 1163, 1165]. We should note [1173], where the usual (in game theory) convexity-type conditions are replaced by the weaker condition of aeyclicness. Here the existence of a saddle point is ensured by a nonzero Lefschetz for an acyclic m-mapping, and the presence of a fixed point (the Eilenberg-Montgomery theorem). Differential games with multivalued payoff functions are considered in [87, 88, 92, 93]. Other applications in the theory of differential games are given in [18, 20, 79, 173,202. 251,325, 1224~ 1225]. Applications to certain other problems in game theory are also considered in [361,866]. Techniques from the theory of multimaps play an important role in research on problems in mathematical economics. Examples of this kind are considered in [47]. The recently translated (into Russian) monograph by Hildebrand [701] contains a detailed description of applications of such parts of the theory of multivalued mappings as fixed-point theorems, measurable m-mappings and their cross-sections, and multivalued integrals in equilibrium theory for mathematical economics. The Kakutani fixed-point theorem is used in [929] to prove a generalization of the Gale-Nikaido-Debreu theorem (see Ref. Zh. Mat., I1V440K, 1972, Theorem 16.6). This result is used to prove the existence of economic equilibrium in generalized Heroux-Debreu models. Here we should also note that Gwinner [654] studied the relationship between a number of fundamental results in nonlinear functional analysis: the Knaster-Kuratowski-Mazurkiewicz theorem, the existence theorem for pseudomonotonic variational inequalities (Ref. Zh. Mat., 11B623, 1969), Fan's minimax principle (see Ref. Zh. Mat., 5B868, 1973), and fixed-point theorems for m-mappings like the Browder, Fan-Glicksberg, and Gale-Nikaido-Debreu theorems. It is shown that all of these results are in some sense equivalent. Multivalued demand functions are investigated in [169]. In addition, we should note A. M. Rubinov's monograph (Ref. Zh. Mat., 5V758K, 1981), which is devoted to superlinear m-mappings, one of the fundamental objects of convex analysis. It considers the asymptotic behavior of" trajectories generated by superlinear mappings defining a model of the Neumarm-Gale dynamics (see, for example, [47]). It also studies the problem of constructing optimal trajectories. Differential inclusions are used in [825] to study main-line trajectories of a functional. Schultz [1108] considers the problem of convex programming in R n with the property that the set of optimal plans in this problem and its dual is nonempty and compact. He studies the problem of continuous relationships between these sets and functions defining the set of feasible plans and goal functions for the initial problem. This problem reduces to investigation of the stability of solutions of some inclusion in R n, for which purpose the topological degree of m-mappings is used. Applications to the stability of parametric linear programming problems are considered in [1157]. Fixed-point theorems for fuzzy m-mappings are applied to economics in [866]. Concerning applications of fuzzy mmappings to economics, see also [344].

w

Generalized Dynamical Systems

Let X be a topological space, and let I" be an additive semigroup with a zero (usually the set of real numbers R). Assume the m-mapping Q: X • r ~ P(X) is such that: 1) Q(., 0) = idx; 2) Q(Q(x, gl), g2) c Q(x, gl + g2) for any x E X; gl, g2 E r define a generalized dynamical system (or a dynamical without uniqueness, or a semidynamical system). Generalized dynamical systems naturally appear in the study of differential equations that do not satisfy uniquesolution conditions, differential inclusions, control systems, in problems on mathematical economics, in problems on the theory of planning, etc. A description of the basic properties of generalized dynamical systems and their trajectories may be found in [46, 47]. Various aspects of the theory of generalized dynamical systems are the subject of an extensive group of investigations. Problems on axiomatization were studied in [264], while conditions for specifying generalized dynamical systems of differential equations and inclusions were considered in [125-129, 943], the properties of funnels in [337], stability

875

in [140, 322, 338, 567, 910], and other properties in [49, 50, 76, 146, 147, 263, 323, 324, 567, 943]. We should note the survey of K. S. Sibirskii [262]. Finding single-valued subsystems and reducing generalized dynamical systems to them was the subject of work by Petukhov [219, 220]. Tsisel'skii [480] investigated continuity properties for generalized dynamical systems inverse to a given system. Application of generalized dynamical systems to problems on dynamic games was studied by V. A. Baidosov [18].

w

Other Applications

Multimaps were applied to the study of spaces with measures in [297, 695, 914]. A series of papers [191, 363, 381, 537-540, 699, 700, 973, 975, 1036, 1071, 1191] has been concerned with applications of m-mappings to research on probability spaces. Application of m-mappings to determination of the relationship between various cones tangent to a subset of a Banach space was described in [992]. A new approach to local analysis of nonsmooth mappings of Banach spaces using m-mappings of a special form was proposed by Ioffe [725]. In [1061] m-mappings are applied to analysis of limit sets characterizing real functions. We should note Novikov's work [186-190] on multivalued functionals. Applications to the theory of decision making may be found in [524, 562]. Opoitsev's monograph [201] applies a fixed-point theorem for m-mappings to research on problems on nonlinear system statics, i.e., problems on system equilibrium, its existence and uniqueness, the relationship to external perturbations, controls, etc. In [581] m-mappings and their fixed points are used in a problem on determining the equilibrium flow in a network. The theory of multivalued mappings is applied to theoretical programming by Smyth [1145]. The notion of semistability for the motion of a point is introduced in [991] to study how the semigroup R+ acts on a metric space. Results were obtained on the connection between semistability and regularity of m-mappings x -, A(x), where A(x) is the limit set of the trajectories of x. In particular, it was shown that semistability of the trajectory of x implies upper semicontinuity of the m-mappings A.

APPENDIX Recent Results This appendix presents a brief survey of the literature on the theory of multivalued mappings for the period 1987-1990. The existence of continuous cross-sections of m-mappings has been the subject of much research. Here we note Michael's paper [62], in which he proves a stronger form of his classical cross-section theorem. A survey by the same author is devoted to the problem of continuous sections that avoid a given set (obstacle). Bressan and Colombo [35] proved an analog of Michael's theorem for an m-mapping with decomposable values (see w Anichini, Conti, and Zecca [30] proved the existence of continuous sections for continuous m-mappings in R n whose ranges are nonempty, compact, contractible, quasiconvex, and locally contractible. The problem of the existence of two continuous sections for a lower semicontinuous mmapping that have no coincident points was considered by Gel'man [13]. The results obtained are applied to proving the existence of convex-valued sections of m-mappings, to fixed-point theorems, and to solvability of the Cauchy problem for differential inclusions with nonconvex right sides. The existence of e-approximations for m-mappings was studied by Anichini [29], where this problem was solved for upper semicontinnous m-mappings F from a locally compact linear metric space into a normed space Y, under the assumption that the values of F are nonempty subsets of Y that are absolute retracts. The existence of e-approximations for uppersemicontinuous m-mappings with decomposable values was proved in [35]. A detailed study of approximable m-mappings was performed by Gorniewicz and Lassonde in the paper "On approximable multi-valued mappings, " which is in press (see also [471). Developing various approaches to differentiability of m-mappings, Rockafellar [68] introduces the notion of protodifferentiability for m-mappings and considers its application to optimization problems. Pham [66] defined the notions of predifferentiation and strict predifferentiation of m-mappings in Banach spaces, and used these notions to establish necessary

876

conditions for optimality in the form of a Lagrange-multiplier rule for problems whose constraints are given in terms of mmappings. Concepts of differentiability, continuous differentiability, and the differential m-mapping were also introduced by Fournier and Violette [40]. They studied certain properties of differentiable m-mappings and applied their results to computing the index of fixed points. New results have also been obtained on multivalued integrals and their applications. M. S. Nikol'skii [23] introduced the notion of KP-integrals of m-mappings. They appear naturally in connection with computing sets of accessibility and their projections in linear control theory. Kandilakis and Papageorgiou [53] have studied the properties of integrals of m-mappings with values in a Banach space, and these results were subsequently used to prove whether control systems are relay systems and in problems on differential inclusions. Artstein [31] investigated parametric integration of m-mappings and considered its application to control systems and stochastic optimization problems. There continues to be considerable interest in research on the topological characteristics of m-mappings, fixed points, and solvability of multivalued operator relations (see [7]). Gel'man [ 11] proved that the structure of the images of boundary points on a ball has no effect on the validity of fixed-point theorems of Kakutani type. In particular if B C R n+I is a bait, S = 0B, and F: B ~ K(Rn+I) is an upper semicontinuous m-mapping such that F(S) C B and for any x E Int B the set F(x) is acyclic, then Fix F ;~ Q, where Fix F = {x EB: x E F(x)}. The set of solutions for inclusions of the form f(x) E F(x) was studied in [10], where conditions were found for this set to be connected or acyclic, and the dimension of this set was investigated. Gel'man and Obukhovskii [14] considered fixed-point theorems for new classes of condensing mappings with convex closed ranges. They proved a proposition that, on the one hand, generalizes the classical Fan-Glicksberg result, and, on the other, a number of fixed-point theorems for condensing m-mappings. This theorem is used to prove an analog of Krasnosel'skii's theorem on fixed points for operators of "condensing plus compact" type. A new class M* of m-mappings including the classes 5z'* and ~* considered in the first parts of this paper (see Sec. 2 in [7] and [411,412] of the bibliography of [7]) was introduced in [33]. Fixed- and coincident-point theorems are proved for this class of m-mappings, and it is noted that an analytic interpretation of these propositions contains a number of wellknown minimax and variational relations. Davidovich [39] introduced a class of spherical m-mappings to which Kakutani fixed-point theorems apply, as well a s the Poincare theorem, the Borsuk-Ulam theorem on antipodes, etc. Chang [37] introduced the notion of fixed points for fuzzy-valued mappings and proved an analog of the Fan-Glicksberg fixed-point theorem. In [48] Hahn proved an analog of the Leray-Schauder fixed-point theorem for random condensing m-mappings. Horvath [49] generalized the Kuratowski-Knaster-Mazurkiewicz theorem, the Fan fixed-point theorem, and minimax inequalities to the case in which the assumption of convexity is replaced by the assumption of contractibility. Development of the Bourgin-Yang theorem is the subject of work by Geba and Gorniewicz [45, 46] and Izydorek [51]. The set {x: F(x) ~ F ( - x ) ~ ~ } is used in [45, 46] to generalize acyclic m-mappings F and multivalued vector fields defined on spheres in finite-dimensional or Banach spaces. An asymmetric version of the Bourgin-Yang theorem is given in [51], which also provides a bound for the dimension of the set {x: ]k > 0, F(x) f) F(-kx)} ;e ~ . Further development in this direction was obtained in [52]. Borisovich [5, 6] presented a systematic study of a group of problems associated with the theory of topological characteristics (rotation, degree, complete Leray index, intersection index) of noncompact m-mappings. Generalizing and unifying a number of earlier constructions of the topological degree, this author studies operator inclusions of the form 0~a (x) - - G (x)

with Fredholm or monotonic operators a and a-condensing multivalued operators G. In the first part of the paper, which deals with the principles of compact condensation and bijective correspondence between homotopic classes, the author establishes the "principle of reduction" of a topological characteristic for the initial inclusion to a characteristic for the inclusion

O~a(x) -If(x) with compact multivalued operator K. The second part of the paper presents concrete examples for general statements for a special class of mappings that are interesting from the viewpoint of application to problems on solvability of nonlinear 877

boundary-value problems and differential inclusions. In particular, a generalized acyclic a-condensing m-mapping G is considered. Kucharski (see [56]) defined a new class of strongly acyclic m-mappings and investigated its properties. An acyclic m-mapping F: X --, K(Y) is said to be strongly acyclic if, for any element x E X, all of the homotopy groups of a nonempty set Y~F(x) are trivial. In the paper "Topological degree theory for acyclic mappings related to the bifurcation problem" by Gorniewicz and Kryszewski, which is in press, there is a definition for a generalized topological degree for strongly acyclic m-mappings and applications are given for investigation of the bifurcation problem for inclusions with acyclic operators. German [12] systematically investigated a topological characteristic of m-mappings with arbitrary compact values in finite-dimensional spaces. He used this invariant to develop a connection between the approximative and homological approaches to the theory of fixed points for m-mappings and proved a number of new fixed-point theorems. Mazur and Werenski [59, 60] constructed a theory for the index of fixed points of (1, a)-bounded and locally acondensing m-mappings with convex values (ix is the Kuratowski noncompactness measure). Gajic [43] defined the topological degree of convex-valued vector fields with uniformly finite-dimensional approximable operators in topological vector spaces. Under certain additional assumptions, the given characteristic is used to define the topological degree of ultimately compact convex-valued vector fields. The topological degree of compact convex-valued vector fields relative to a convex closed subset of a topological vector space was defined in [44]. Developing results of Browder (Ref. Zh. Mat., 10B885, 1983; 12B1217, 1983), Zhang and Chen provide a construction for the topological degree of m-mappings of types (S) and (S)+, describe the fundamental properties of the degree, and consider applications to solvability of operator inclusions and fixed-point theorems. Bogatyi [3] studied the fixed-point index (in the sense of Zigberg and Skordev; see [1127] of [7]) for m-mappings of compact polyhedra into themselves that are composites of acyclic mappings. A general construction for the index of fixed points of acyclic m-mappings on ENR's, which combines a number of earlier results, was proposed by Bielawski [34]. The notion of differentiability for m-mappings is used in [40] to define the index of fixed points of m-mappings in a Banach space that are composites of acyclic m-mappings. Skordev [71] proved that the fixed-point index of the product of two acyclic m-mappings of a metric ANR-space is equal to the product of the indexes of the factors. Different forms of the Lefschetz fixed-point theorem for acyclic mhaappings were considered in [70]. The Lefschetz number and fixed-point index for m-mappings approximable by single-valued mappings was studied by Gorniewicz, Grams, and Kryszewski [47]. Obukhovskii [27] considered the following form of the theory of topological degrees of m-mappings: Let E I and E 2 be locally convex spaces, and assume that U C E 1 is open. Let .~z'r: be some class of m-mappings H: E 1 --, P(E2) such that supp H = {x E El: H(x) ;~ {0}} C U. An upper semicontinuous m-mapping ~: [1 --, K(E2) is said to be Sty-essential if 9 -1(0) C Uandforany /-/6~r~: there exists a coincident point x o E U:~(Xo) N H(xo) ;~ ~ . For the ease in which Wt:=Wt7 is a set of compact m-mappings, he describes the properties of W~T -essential m-mappings, provides applications to theorems on coincident points of m-mappings, solvability of abstract boundary-value problems, and investigation of the topological structures of sets of solutions for relations with essential multivalued operators. Kryszewski and Miklaszewsld [54] considered m-mappings F whose ranges satisfy the following condition: For each > 0 there exists a 6, 0 < ~ < e, such that an imbedding of the neighborhood Ut(F(x)) in the neighborhood Ue(F(x)) generates trivial homomorphisms for all homotopy groups. The method of single-valued approximations is used to define the Nielsen number of fixed points for composites of m-mappings of the indicated class into compact ANR-spaces. We should note that the first to prove the existence of single-valued approximations for practically the same class of aspherical mmappings was A. D. Myshkis (see Ref. Zh. Mat., 732, 1955). Borisovich and Gliklikh (Ref. Zh. Mat., 1A821, 1972) defined the Lefschetz number for the indicated class of m-mappings, and Gliklikh (Ref. Zh. Mat., 2B896, 1973) constructed the rotation of the corresponding multivalued vector fields (see also the survey by Borisovich, Gel'man, Myshkis, and Obukhovskii (Ref. Zh. Mat., 6A571, 1980)). In recent years the theory of multivalued mappings has been applied to a very diverse group of problems. One of the most productive areas has been the theory of differential inclusions and the theory of optimization. We would like to call attention to the following work. In the theory of differential inclusions, there has been extensive development of the theory of differential inclusions in Banach spaces. Papageorgiou [65] and Obukhovskii [63] studied the existence of weak solutions to semilinear differential 878

evolution inclusions in a separable Banach space E of the form

x' (0 ~A (t)x (t) + F (t, x(t) ), t~[0, T], where {A(0}t~10,TI is a family of closed, linear, not necessarily bounded operators in E, F is a convex-valued m-mapping satisfying Caratheodory-type conditions and a regularity condition of the type

X(F(t, O) )<~k(t)x(D) for any bounded D C E, where k E L+I([0, T]), and X is the Hausdorff noncompactness measure in E. An existence theorem for weak solutions of functional-differential inclusions in Banach spaces of the form

x" (t) ~A (t) x (t) + E (t. x,), where xt(0) = x(t + 0), 0 E ]-~', 0], was proved by Obukhovskii in [26]. In this paper he also investigated the properties of solutions and applications to optimization problems for parabolic control systems with delays. Semilinear differential inclusions in Banach spaces were investigated by Frankowska [42], who cousidered the relationship between the trajectories of the initial and convex inclusions and obtained an equation for the set of accessibility. The results were applied to investigation of an optimal control problem in Banach spaces. Boundary-value problems for semilinear differential evolution inclusions in Banach spaces were considered in [64]. The existence of a local solution to the Cauchy problem in a Banach space

x'(O~F(x(O), x(O)=xo under the condition that the m-mapping F is continuous and ~F(xo) is of finite codimension was proved by Bressan and Colombo [36]. Kubiaczyk [55] proved the existence of solutions for hyperbolic differential inclusions in a Banach space. Differential inclusions with fuzzy right sides were considered by Baidosov [1]. The existence of periodic solutions for semilinear differential inclusions in R n was studied by Macki, Nistri, and Zecca [58]. The theory of topological degrees of condensing m-mappings was applied by Kamenskii and Obukhovskii [17] to. investigate problems on periodic solutions of differential inclusions in Banach spaces of the form

x' (t) ~Ax (t) +F(t, x (t) ), where A is an unbounded and generally linear operator. They considered applications to problems on existence of optimal periodic modes for control systems in Banach spaces. A detailed paper, "On periodic solutions of differential inclusions with unbounded operators in Banach spaces, " by Kamenskii and Obukhovskii is to be published by Zbornik Radova (Yugoslavia). Klimov [19] used methods from the theory of topological degrees of m-mappings to investigate periodic problems of the form y' E Ay, y(0) = y(T), where A is a parabolic operator between functional spaces. Rotation of the multivalued vector field generated by the operator A is used to prove solvability for a periodic problem, and the method of averaging is used to consider a form of similarity principle and Bogolyubov's second theorem. Bogatyrev [4] considered differential inclusions in which the right side contains a dense set of continuous sections that are Lipschitzian with unit constant. A fixed-point theorem'for m-mappings with decomposable values is used to prove that the family of solutions for the inclusion is connected. Kurzhanski and Filippova [57] considered problems associated with evolution, estimation, and control of dynamic processes under conditions of indefiniteness described by differential inclusions. Nikol'skii [24] obtained differential inclusions in variations that are generalizations of equations in variations known in the smooth case for ordinary differential vector equations with nonsmooth right sides depending on a parameter. Gliklikh [15] obtained a condition for accessibility of certain submanifolds by the solutions of second-order differential equations in manifolds that have a mechanical interpretation in terms of Newton's laws for the case of explosive forces in the presence of mechanical coupling. Clarke [18] investigated the theory of the generalized gradient he defined and investigated optimization problems for control systems described by differential inclusions: He studied conditions for controllability, proved a very general form of the principle of the maximum, and presented sufficient conditions for optimality. Among the applications discussed are a variational proof of Aumann's theorem on convexity of multivalued integrals. 879

Necessary conditions for optimality and controllability of systems described by differential inclusions with delays were considered by Clarke and Watldns in [38]. Morduldaovich [22] applied the theory of multivalued mappings and differential inclusions to developing methods for solving complex optimization and control problems with nonsmooth and nonconvex structures. Borisovich and Obuldaovskii [8, 9] used theorems on existence of solutions to differential inclusions in Banach spaces to investigate optimization problems for parabolic control systems with feedback. For example, they solve an optimization problem for controlling thermal propagation in isotropic solids. A similar problem for control systems with delays was considered by Obukhovsldi in [26]. Problems on controllability and optimization of control systems were investigated by Schilling [69] by means of algorithms using fixed points of m-mappings. Frankowska [41] used a form of Pontryagin's principle of the maximum to obtain necessary conditions for optimality for differential inclusions with end-point constraints. Problems on optimization of systems described by evolutions inclusions Mel'nlk [21] and Ivanenko and Mel'nik [16]. Aubin and Frankowska [32] applied a theorem on inverse functions for m-mappings to derivation of sufficient conditions for local uniqueness of solutions to inclusions, investigation of problems on nonsmooth optimization, and the study of local controllability of differential inclusions. Pshenichny [67] presented a variety of implicit-function theorems for m-mappings that are applicable to solution of problems in mathematical programming. A new area in functional analysis -- subdifferential calculus -- is the subject of a monograph by Kursaev and Kutateladze [20] and work by Ioffe [50]. The theory of subdifferentials and generalized gradients is also considered in Aubin's monograph [25]. This same monograph presents a detailed description of contemporary approaches to applying the theory of multivalued mappings to game theory and mathematical economics. N. V. Senchakova (see [28]) used the notion of rotation of multivalued vector fields to investigate solvability of variational inequalities. Broad application of the theory of multivalued mappings in research on variational and quasivariational inequalities is described by Baiocchi and Capelo [2]. REFERENCES 1. 2. 3. 4. 5. 6. 7.

8.

9.

10. 11.

880

V.A. Baidosov, "Fuzzy differential inclusions," Prikl. Mat. Mekh., 54, No. 1, 12-17 (1990). C. Baiocci and A. Capelo, Variational and Quasivariational Inequalties: Applications to Problems With Free Boundaries [Russian translation from Italian], Nauka, Moscow (1988). S.A. Bogatyi, "Indices of iterations of multimaps," Dold. Bolg. AN, 41, No. 2, 13-16 (1988). A.V. Bogatyrev, "Fixed points and autonomous differential inclusions," Differents. Uravnen., 22, No. 8, 1298-1308 (1986). Yu. G. Borisovich, "A modern approach to the theory of topological characteristics of nonlinear operators. I.," in: Geom. and Singularity Theory for Nonlinear Equats. [in Russian], Voronezh (1987), pp. 24-46. Yu. G. Borisovich, "A modern approach to the theory of topological characteristics of nonlinear operators. II.," in: Global Analysis and Nonlinear Equations [in Russian], Voronezh (1988), pp. 22-43. Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, and V. V. Obukhovskii, "New results in the theory of multivalued mappings. I. Topological characteristics and solvability of operator relations," Itogi Nauki i Tekhn. Ser. Mat. Anal., 25, 123-197 (1987). Yu. G. Borisovich and V. V. Obukhovskii, "Problems on optimizing parabolic control systems with feedback," in: Abstracts of the Third All-Union Schools on "Pontryagin Readings" [in Russianl, Kemerovo (1990), p. 113. Yu. G. Borisovich and V. V. Obukhovskii, "An optimization problem for control systems described by parabolic differential equations," in: Algebraic Problems in Analysis and Topology [in Russian], Voronezh (1990), pp. 105-109. B . D . Gel'man, "The structure of sets of solutions to inclusions with multivalued operators," in: Global Analysis and Mathematical Physics [in Russian], Voronezh (1987), pp. 26-41. B.D. Gel'man, "Fixed-point theorems of Kakutani type for multimaps," in: Global Analysis and Nonlinear Equations [in Russian], Voronezh (1988), pp. 117-119.

12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

B. D. Gel'man, "Problems on the theory of fixed points of multimaps," in: Topology and Geometric Methods of Analysis [in Russian], Voronezh (1989), pp. 90-105. B. D. Gel'man, "Continuous selections of multimaps without coincident points," Pr. Uniw. Gdaftski. Inst. mat., No. 74, 1-18 (1990). B. D. Gel'man, "Fixed-point theorems for condensing multimaps," Algebraic Problems of Analysis and Topology [in Russian], Voronezh (1990), pp. 110-115. Yu. E. Gliklilda, "Integral operators and differential inclusions on manifolds subject to noninvolutive distributions," in: Issues in Analysis and Differential Topology [in Russian], Kiev (1988), pp. 22-28. V. I. Ivanenko and V. S. Mel'nik, "Optimal control by evolution equations with multivalued operators," Dold. Akad. Nauk SSSR, 297, No. 1, 26-30 (1987). M. I. Kamenskii and V. V. Obukhovskii, "Condensing operators and problems on periodic solutions for parabolic differential inclusions," in: Abstracts of the Fourteenth Schools on the Theory of Operators in Functional Spaces [in Russian], Vol. 2, Novgorod (1989), p. 3. F. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York (1983). V. S. Klimov, "The problem of periodic solutions of differential operator inclusions," Izv. Akad. Nauk. SSSR, Ser. Mat., 53, No. 2, 309-327 (1989). A. G. Kursaev and S. S. Kutateladze, Subdifferential Calculus [in Russian], Nauka, Novosibirsk (1987). V. S. Mel'nik, "Optimal control by evolution equations containing multivalued operators," Kibernetika, No. 5, 66-70 (1986). B. Sh. Mordukhovich, Approximation Methods for Optimization Problems and Control [in Russian], Nauka, Moscow (1988). M. S. Nikol'skii, "KP-integrals for multimaps," Differents. Uravnen., 23, No. 10, 1705-1710 (1987). M. S. Nikol'skii, "Differential inclusions in variations for differential equations with nonsmooth right side," Dokl. po. Mat. i Ee Pril. Akad. Nauk SSSR Mat. Inst. Tul. Politekh. Inst., 2, No. 2, 197-207 (1988). J.-P. Aubin, Nonlinear Analysis and Its Economic Applications [Russian translation from French], Mir, Moscow (1988). V. V. Obukhovskii, "Semilinear functional-differential inclusions in Banach spaces and parabolic control systems," Avtomatika, No. 3 (1991). V. V. Obuldaovsldi, "Essential mulfimaps," in: Proc. Baku International Topol. Conf. [in Russian], Baku (1989), pp. 137-142. N. V. Senchakova, "Rotation of multivalued vector fields and solution of variational inequalities," in: Qualitative Methods for Operator Equations [in Russian], Yaroslval' (1988). G. Anichini, "Approximate selections for nonconvex set valued mapping," Boll. Unione Mat. Ital., B(7)4, No. 2, 313-326 (1990). G. Anichini, G. Conti, and P. Zecca, "Un teorema di selezione per applicazioni multivoche avalorinon convessi," Boll. Unlone Mat. Ital., C5, No. 1,315-320 (1986). Z. Artstein, "Parametrized integration of multifunctions with applications to control and optimization," SIAM J. Contr. and Optim., 27, No. 6, 1369-1380 (1989). J.-P. Aubin and H. Frankowska, "On inverse function theorems for set-valued maps," J. Math. Pures et Appl., 65, No. 1, 71-89 (1987). H. Ben-EI-Mecha'/ekh, P. Deguire, and A. Granas, "Points fixes et coincidences pour les fonctions multivoques. III (Application de type M* et M)," Compt. Rend. Acad. Sci. Str. I, 305, No. 9, 381-384 (1988). R. Bielawsld, "The fixed point index for acyclic maps on ENR-s," Bull. Pol. Acad. Sci. Math., 35, Nos. 78, 487-499 (1987). A. Bressan and G. Colombo, "Extensions and selections of maps with decomposable values," Stud. Math., 90, No. 1, 69-86 (1988). A. Bressan and G. Colombo, "Generalized Baire category and differential inclusions in Banach spaces," J. Different. Equat., 76, No. 1, 135-158 (1988). Chang Shih-sen, "Fixed degree for fuzzy mappings and a generalization of Ky Fan's theorem," Fuzzy Sets and Syst., 24, No. 1, 103-112 (1987).

881

38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

48. 49.

50. 51. 52. 53. 54. 55. 56. 57.

58. 59. 60. 61. 62.

882

F. H. Clarke and G. G. Watldns, "Necessary conditions, controllability and the value function for differential~lifferenceinclusions," Nonlinear Anal. Theory, Meth. and Appl., 10, No. 11, 1155-1179 (1986). A. Dawidowicz, "Spherical maps," Fundam. Math., 127, No. 3, 187-196 (1987). G. Fournier and D. Violette, "A fixed point theorem for a class of multi-valued continuously differentiable maps," Ann. Pol. Math., 47, No. 3, 381-402 (1987). H. Frankowska, "The maximum principle for an optimal solution to a differential inclusion with end points constraints," SIAM J. Contr. and Optim., 25, No. 1, 145-157 (1987). H. Frankowska, "Estimations a priori pour les inclusions differentielles op6rationnelles," Compt. Rend. Acad. Sci. S6r. 1,308, No. 2, 47-50 (1989). Lj. G-aji6, "Topological degree for ultimately compact multivalued vector fields in topological vector spaces," Mat. Vesn., 39, No. 2, 145-158 (1987). Lj. Gajid, "On relative topological degree of set-valued compact vector fields," Indian J. Pure and Appl. Math., 19, No. 3,269-276 (1988). K. G~ba and L. G6rniewicz, "On the Bourgin-Yang theorem for multi-valued maps. I," Bull. Pol. Acad. Sci. Math., 34, Nos. 5-6, 315-322 (1986). K. G~ba and L. G6rniewicz, "On the Bourgin-Yang theorem for multivalued maps. II," Bull. Pol. Acad. Sci. Math., 34, Nos. 5-6, 323-329 (1986). L. G6rniewicz, A. Grams, and W. Kryszewski, "Sur la m6thode de l'homotopie dans la th6orie des points fixes pour les applications multivoques. Pattie 2. L'indice dans les ANR-s compacts," Compt. Rend. Acad. Sci. S6r. 1,308, No. 14, 449-452 (1989). S. Hahn, "A general random fixed point theorem for upper semicontinuous multivalued 1-set-contractions," Z. Anal. und. Anwend, 6, No. 3, 281-286 (1987). Ch. Horvath, "Some results on multivalued mappings and inequalities without convexity," in: Proc. Sem. on Nonlinear and Convex Anal. in Hon. of Ky Fan, Santa Barbara, June 23-26, 1985, New York (1987), pp. 99-106. A. D. Ioffe, "On the theory of subdifferential," in: FERMAT Days 85: Math. Optimiz., Amsterdam (1986), pp. 183-200. M. Izydorek, "On the Bourgin-Yang theorem for mult-valued maps i n the non-symmetric case. II," Seredika. B"lt. Mat. Spis., 13, No. 4, 420-422 (1987). M. Izydorek, "Remarks on Borsuk-Ulam theorem formulti-valuedmaps," Bull. Pol. Acad. Sci. Math., 35, Nos. 7-8, 501-504 (1987). D. A. Kandilakis and N. S. Papageorgiou, "On the properties of the Aumann integral with applications to differential inclusions and control systems," Czechosl. Math. J., 39, No. 1, 1-15 (1989). W. Kryszewski and D. Miklaszewski, "The Nielsen number of set-valued maps. An Approximation approach," Serdika B"lg. Mat. Spis., 15, No. 4, 336-344 (1989). I. Kubiaczyk, "Existence theorem for multivalued hyperbolic equation," Rocz. PTM. Set. 1, 27, No. 1, 115119 (1987). Z. Kucharski, "On a class of strongly acyclic maps," Serdika. B"lg. Mat. Spis., 14, No. 2, 195-197 (1988). A. B. Kurzhanksi and T. F. Filippova, "On the set-valued calculus in problems of viability and control for dynamic processes: the evolution equation," Ann. Inst. H. Poincar6. Ann. Nonlineaire, 6, Suppl., 339-363 (1989). J. W. Maeki, P. Nistri, and P. Zecca, "The existence of periodic solutions to nonautonomous differential inclusions," Proc. Am. Math. Soc., 1(}4, No. 3, 840-844 (1988). T. Mazur and S. Were~ki, "The topological degree and fixed point theorem for 1-set contractions," Ann. UMCS. Sec. A: Math., 38, 79-84 (1984). T. Mazur and S. Werefiski, "The degree theory for local condensing mappings," Ann. UMCS. Sec. A: Math., 38, 85-90 (1984). E. Michael, "Continuous selections: A guide for avoiding obstacles," in: Proc. Sixth Prague Topol. Symp. Aug. 25-29, 1986, Berlin (1988). E. Michael, "A generalization of a theorem on continuous selections," Proc. Am. Math. Sot., 105, No. 1, 236-243 (1989).

63. 64. 65. 66. 67. 68. 69. 70. 71. 72.

V. Obukhovsldi, "On evolution differential inclusions in a Banach space," in: Proc. Fourth Conf. Diff. Equations and Their App., Ruse (1989), p. 430. N. S. Papageorgiou, "Boundary value problems for evolution inclusions," Comment. Math. Univ. Carol., 29, No. 2, 355-363 (1988). N. S. Papageorgiou, "On multivalued semilinear evolution equations," Boll. Unione Mat. Ital. B., 3, No. 1, 1-16 (1989). Pham Huu Sach, "Differentiability of set-valued maps in Banach spaces," Math. Nachr., 139, 215-235 (1988). B. N. Pshenichny, "Implicit function theorems for muhi-valued mappings," in: Proc. Fourth Course Int. School Math., Erice, June 20-July 1, 1988, New York-London (1989). R. T. Rockafellar, "Protodifferentiability of set-valued mappings and its applications in optimization," Ann. Inst. H. Poincare. Anal. Nonlineaire, 6, Suppl., 449-482 (1989). K. Schilling, "An approach to control theory by fixed point algorithm," Lect. Notes Contr. and Inf. Sci., 95, 56-78 (1987). G. S. Skordev, "The Lefschetz fixed point theorem," God. Sofiisk. Univ. Fak. Mat. i Mekh. Mat., 79, No. 1,201-213 (1989). G. S. Skordev, "The multiplicativity property of the fixed point index for multivalued maps," Serdika. B"lg. Mat. Spis., 15, No. 2, 160-170 (1989). Zhang Shi-sheng and Chen Yu-chin, "Degree theory for multivalued (S) type mappings and fixed point theorems," Appl. Math. and Mech., 11, No. 5, 441-454 (1990).

883